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1.1.6: Scaling and Area - Mathematics

1.1.6: Scaling and Area - Mathematics



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Lesson

Let's build scaled shapes and investigate their areas.

Exercise (PageIndex{1}): Scaling a Pattern Block

Use the applets to explore the pattern blocks. Work with your group to build the scaled copies described in each question.

  1. How many blue rhombus blocks does it take to build a scaled copy of Figure A:
    1. Where each side is twice as long?
    2. Where each side is 3 times as long?
    3. Where each side is 4 times as long?
  2. How many green triangle blocks does it take to build a scaled copy of Figure B:
    1. Where each side is twice as long?
    2. Where each side is 3 times as long?
    3. Using a scale factor of 4?
  3. How many red trapezoid blocks does it take to build a scaled copy of Figure C:
    1. Using a scale factor of 2?
    2. Using a scale factor of 3?
    3. Using a scale factor of 4?
  4. Make a prediction: How many blocks would it take to build scaled copies of these shapes using a scale factor of 5? Using a scale factor of 6? Be prepared to explain your reasoning.

Exercise (PageIndex{2}): Scaling More Pattern Blocks

Your teacher will assign your group one of these figures, each made with original-size blocks.

  1. In the applet, move the slider to see a scaled copy of your assigned shape, using a scale factor of 2. Use the original-size blocks to build a figure to match it. How many blocks did it take?
  2. Your classmate thinks that the scaled copies in the previous problem will each take 4 blocks to build. Do you agree or disagree? Explain you reasoning.
  3. Move the slider to see a scaled copy of your assigned shape using a scale factor of 3. Start building a figure with the original-size blocks to match it. Stop when you can tell for sure how many blocks it would take. Record your answer.
  4. Predict: How many blocks would it take to build scaled copies using scale factors 4, 5, and 6? Explain or show your reasoning.
  5. How is the pattern in this activity the same as the pattern you saw in the previous activity? How is it different?

Are you ready for more?

  1. How many blocks do you think it would take to build a scaled copy of one yellow hexagon where each side is twice as long? Three times as long?
  2. Figure out a way to build these scaled copies.
  3. Do you see a pattern for the number of blocks used to build these scaled copies? Explain your reasoning.

Exercise (PageIndex{3}): Area of Scaled Parallelograms and Triangles

  1. Your teacher will give you a figure with measurements in centimeters. What is the area of your figure? How do you know?
  2. Work with your partner to draw scaled copies of your figure, using each scale factor in the table. Complete the table with the measurements of your scaled copies.
    scale factorbase (cm)height (cm)area (cm2)
    (1)
    (2)
    (3)
    (frac{1}{2})
    (frac{1}{3})
    Table (PageIndex{1})
  3. Compare your results with a group that worked with a different figure. What is the same about your answers? What is different?
  4. If you drew scaled copies of your figure with the following scale factors, what would their areas be? Discuss your thinking. If you disagree, work to reach an agreement. Be prepared to explain your reasoning.
scale factorarea (cm2)
(5)
(frac{3}{5})
Table (PageIndex{2})

Summary

Scaling affects lengths and areas differently. When we make a scaled copy, all original lengths are multiplied by the scale factor. If we make a copy of a rectangle with side lengths 2 units and 4 units using a scale factor of 3, the side lengths of the copy will be 6 units and 12 units, because (2cdot 3=6) and (4cdot 3=12).

The area of the copy, however, changes by a factor of (scale factor)2. If each side length of the copy is 3 times longer than the original side length, then the area of the copy will be 9 times the area of the original, because (3cdot 3), or (3^{2}), equals 9.

In this example, the area of the original rectangle is 8 units2 and the area of the scaled copy is 72 units2, because (9cdot 8=72). We can see that the large rectangle is covered by 9 copies of the small rectangle, without gaps or overlaps. We can also verify this by multiplying the side lengths of the large rectangle: (6cdot 12=72).

Lengths are one-dimensional, so in a scaled copy, they change by the scale factor. Area is two-dimensional, so it changes by the square of the scale factor. We can see this is true for a rectangle with length (l) and width (w). If we scale the rectangle by a scale factor of (s), we get a rectangle with length (scdot l) and width (scdot w). The area of the scaled rectangle is (A=(scdot l)cdot (scdot w)), so (A=(s^{2})cdot (lcdot w)). The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two-dimensional figures too, not just for rectangles.

Glossary Entries

Definition: Area

Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.

For example, the area of region A is 8 square units. The area of the shaded region of B is (frac{1}{2}) square unit.

Definition: Corresponding

When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.

For example, point (B) in the first triangle corresponds to point (E) in the second triangle. Segment (AC) corresponds to segment (DF).

Definition: Reciprocal

Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is (frac{1}{12}), and the reciprocal of (frac{2}{5}) is (frac{5}{2}).

Definition: Scale Factor

To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.

In this example, the scale factor is 1.5, because (4cdot (1.5)=6), (5cdot (1.5)=7.5), and (6cdot (1.5)=9).

Definition: Scaled Copy

A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.

For example, triangle (DEF) is a scaled copy of triangle (ABC). Each side length on triangle (ABC) was multiplied by 1.5 to get the corresponding side length on triangle (DEF).

Practice

Exercise (PageIndex{4})

On the grid, draw a scaled copy of Polygon Q using a scale factor of 2. Compare the perimeter and area of the new polygon to those of Q.

Exercise (PageIndex{5})

A right triangle has an area of 36 square units.

If you draw scaled copies of this triangle using the scale factors in the table, what will the areas of these scaled copies be? Explain or show your reasoning.

scale factor area (units2)
(1)(36)
(2)
(3)
(5)
(frac{1}{2})
(frac{2}{3})
Table (PageIndex{3})

Exercise (PageIndex{6})

Diego drew a scaled version of a Polygon P and labeled it Q.

If the area of Polygon P is 72 square units, what scale factor did Diego use to go from P to Q? Explain your reasoning.

Exercise (PageIndex{7})

Here is an unlabeled polygon, along with its scaled copies Polygons A–D. For each copy, determine the scale factor. Explain how you know.

(From Unit 1.1.2)

Exercise (PageIndex{8})

Solve each equation mentally.

  1. (frac{1}{7}cdot x=1)
  2. (xcdotfrac{1}{11}=1)
  3. (1divfrac{1}{5}=x)

(From Unit 1.1.5)


Area & Perimeter of a Rectangle Calculator

Objective :
Find what is area of the rectangle for given input data?

Formula :
Area = length x width

Solution :
Area = 5 x 10
Area = 50 in²

  1. Enter the length and width of a rectangle in the box. These values must be positive real numbers or parameter. Note that the length of a segment is always positive
  2. Press the "GENERATE WORK" button to make the computation
  3. Rectangle calculator will give the perimeter, area and diagonal length of a rectangle.

where $a$ and $b$ are the length and width of the rectangle, respectively.

Area of Rectangle Formula: The area of a rectangle is determined by the following formula

where $a$ and $b$ are the length and width of the rectangle, respectively.

Length of Diagonal of Rectangle Formula: The diagonal of a rectangle is determined by the following formula


6.3 Area of Scaled Parallelograms and Triangles

  1. Your teacher will give you a figure with measurements in centimeters. What is the area of your figure? How do you know?
  2. Work with your partner to draw scaled copies of your figure, using each scale factor in the table. Complete the table with the measurements of your scaled copies.
    scale factorbase (cm)height (cm)area (cm 2 )
    1
    2
    3
    frac
    frac
  3. Compare your results with a group that worked with a different figure. What is the same about your answers? What is different?
  4. If you drew scaled copies of your figure with the following scale factors, what would their areas be? Discuss your thinking. If you disagree, work to reach an agreement. Be prepared to explain your reasoning.
    scale factorarea (cm 2 )
    5
    frac

Basic Multiplication

It's convenient to think of the logarithm as the common (base 10) logarithm, and the length of the slide rule as one unit, but you can also think of log meaning the natural logarithm, and the length of the slide rule being log(10) units.

  1. The real number line is infinite and slide rules have finite length. Hence all scales can only show a part of the real number line. On the C and D scales, any number x is shown as a number between 1 and 10, and it is determined only up to a factor that is an integer power of 10. In other words your slide rule does not usually show the location of the decimal point. You are supposed to understand your problem well enough so you can tell where to put it. The slide rule also does not tell you the sign of your result.
  2. Compared to a calculator, a slide rule is severely limited in its accuracy. You can enter and read a number typically to two or three decimal digits only.

Scales

All other scales on a slide rule are referenced to the C and D scales. Following is a list of scales commonly found on slide rules. For each scale we list the name (like C), the function underlying it (like ), and some explanations or comments.

CI, DI CI is on the slide, DI on the body.

CF, DF CF is on the slide, DF on the body.

CIF, DIF CIF is on the slide, DIF on the body.

A, B A is on the body, B is on the slide.

R , W May come with subscripts to distinguish and , and have a prime attached to distinguish location on the body or slide. These scales are labeled R (Root) or W (Wurzel). The radical symbol may also be used.

K This scale usually occurs by itself, rather than as a member of a pair. LL, E or This is one of the scales that show the decimal point. Usually there are several scales, like

L The only scale on a slide rule that has a constant increment. Usually on the slide. If there was one such scale on the slide and one on the body they could be used for the addition of numbers.

S , Lists the angle for which of . On slide rules, all angles are measured in degrees, and reside in the interval . The scale usually lists both and , using the identity

T , Similar to the S scale. is in the interval , is in and . There may be a similar scale of in the interval in which case subscripts may be used to distinguish the scales.

ST showing the angle (in degrees) in the unit circle for an arc of length where is in the interval . For such small arcs, within the accuracy of a slide rule, the angle (measured in radians), the sine, and the tangent are all equal.

P for in the interval . The Pythagorean Scale.

H for in the interval . There may be another scale for in and the two scales may be distinguished by subscripts.

Sh is the inverse of the hyperbolic sine. is in the interval If a scale is present for in the scales may be distinguished by subscripts.

Ch is the inverse of the hyperbolic cosine. is in the interval .

Th is the inverse of the hyperbolic tangent. is in the interval .

Table 1: Common Scales

One Variable

More generally, if you choose a number on a scale corresponding to the function (as listed in Table 1), and you read the corresponding number on a scale corresponding to the function , then

where is the inverse function of . The rows of tables 2 and 3 correspond to , and the columns to .


Note that is not the number under the hairline on the C scale, unless you choose to start on that scale!

Table 2: One Variable Conversion

Table 3: More One Variable Conversion

There are some caveats about reading Tables 2 and 3. For example, may have to be in a certain interval, and the tables do not distinguish between different versions of the same scale, e.g., the various LL scales. For the S scale, we only consider the inverse sine function, not the inverse cosine function. So before you use your slide rule as suggested in the tables you'll have to think carefully about what you are doing, which never hurts anyway. The typesetting of some of those formulas is a bit idiosyncratic. They were mostly machine generated, and I did not want to introduce additional errors by excessive manual editing.

As the tables clearly indicate, if you move the hairline over any number on any scale at all, and read the number on the same scale right under the hairline, you'll get that very same number back!

Two Variables

Of course the number of possibilities is vastly increased by allowing the slide to move. We consider two procedures, PLUS and MINUS, involving scales 1, 2, and 3. Scales 1 and 3 are on the body, scale 2 is on the slide.

PLUS: Select u on scale 1 (on the body), align it with the index of scale 2 (on the slide), move the hairline to v on scale 2, and read the result on scale 3 (on the body), underneath the hairline. For example if the scales involved are D , C , and D , the result would be the product, uv .

MINUS: Select u on scale 1, align it with v on scale 2 on the slide, move the hairline to the index of scale 2, and read the result on scale 3 on the body, underneath the hairline. For example, if the scales involved are again D , C , and D , the result is the quotient, .

What happens if we use other scales? Assuming a (very hypothetical) slide rule that has all the scales listed above both on the body and on the slide, these two procedures let you evaluate 3,540 different expressions in 4,394 different ways. Six examples are given in Table 4. Click here to see a similarly organized pdf file (of several hundred pages) showing all the possibilities.

In general, if is the function corresponding to scale 1 (again, as listed in Table 1), the function corresponding to scale 2, and the function corresponding to scale 3, then the result that you read on scale 3 is

where the base of the logarithm is the length of the slide rule and exp is the inverse function of log. The symbol indicates whether to use the plus or the minus procedure.

row entry formula variation result Scale 1 Scale 2 Scale 3 +/-
1 1 1 1 CD CD CD +
2 15 2 1 CD CD CD -
3 2403 1803 1 LL CD LL +
4 139 26 2 CD CDI H +
5 287 83 1 CD AB W -
6 424 168 1 CD LL S -

Table 4: Two Variable Computations

The first three rows of Table 4 show the most common operations on a slide rule: product, quotient, and power.

The last three rows show less common formulas that can be evaluated. Thus, according to the fourth row, to compute follow the PLUS procedure with scales 1, 2, and 3 being D , CI , and H , respectively. The first number in that row, 139, indicates the entry in the pdf table, 26 means it is the 26th distinct formula in the table, and 2 means it's the second way to evaluate this particular formula. These numbers are not important for the example, but they illustrate the organization of the pdf table. Caveats apply even more so than to the one variable Table 2 and 3 above. The variables have to be in certain ranges, and you may have to be judicious about which variant of the relevant scale you use to read your result.

Of course, slide rule manuals do not list thousands of formulas. They describe basic principles and then people can figure out how to use slide rules to best advantage for their particular applications. There are more pedestrian ways to compute but if you have to evaluate such expressions many times you'll find the shortcut eventually. Once you have it you can impress your friends and coworkers!

The last example in Table 4 requires an LL scale on the slide. When I went to high school our work horse slide rule was the Aristo Scholar 903. One version of it has a body and cursor with one side, but a slide with two sides. The back of the slide shows several LL scales. So prior to doing this calculation you need to turn the slide around. This gives you a very strange slide rule without a C scale. For years I have wondered for what kind of application one would want to turn the slide on the Aristo Scholar, and after writing this web page I know!

Three Variables

With the 13 scales assumed here, there are 24,314 distinct such expressions, filling 2,143 printed pages that you can view or download here. The four columns following the mathematical expression give the scales 1, 2, 3, and 4 being used.

Sophisticated Multiplication and Division

Quadratic Equations

As discussed above, one thing slide rules can do that calculators can't is create tables. Here is an intriguing application of that idea that I found in the Post Versalog Slide Rule Instructions, Frederick Post Company, 1963. That readable little book describes very many applications of slide rules.

Suppose we want to find the roots of the equation

Let's assume that is positive, and the roots are real. If is negative we ignore that fact and worry about the signs of the solutions later. As an exercise you may want to figure out what happens when the roots of the quadratic equation are complex. If the solutions are and we have

So we want to find two numbers and that add to and multiply to . We move the hairline over on the D scale, and place the beginning or end of the slide under the hairline (choosing whichever causes the smaller projection of the slide beyond the body). Now the product of any pair of numbers on the D and CI scales (or on the DF and CIF scales) is equal to . Your slide rule now contains a table of pairs of numbers that all have the same product. All that's left to do is to move the hairline until we find a pair of numbers on the D and CI scales (or DF and CIF scales) that add to . Computing the sums mentally as we move the hairline is a pleasant exercise that requires no external help. Once we have the pair of numbers we can figure out the sign of the roots from the signs of and .


Free Math Worksheets for Grade 6

This is a comprehensive collection of free printable math worksheets for sixth grade, organized by topics such as multiplication, division, exponents, place value, algebraic thinking, decimals, measurement units, ratio, percent, prime factorization, GCF, LCM, fractions, integers, and geometry. They are randomly generated, printable from your browser, and include the answer key. The worksheets support any sixth grade math program, but go especially well with IXL's 6th grade math curriculum.

The worksheets are randomly generated each time you click on the links below. You can also get a new, different one just by refreshing the page in your browser (press F5).

You can print them directly from your browser window, but first check how it looks like in the "Print Preview". If the worksheet does not fit the page, adjust the margins, header, and footer in the Page Setup settings of your browser. Another option is to adjust the "scale" to 95% or 90% in the Print Preview. Some browsers and printers have "Print to fit" option, which will automatically scale the worksheet to fit the printable area.

All worksheets come with an answer key placed on the 2nd page of the file.

Multiplication and Division and Some Review

  • 1-digit divisor, 5-digit dividend, no remainder
  • 1-digit divisor, 5-digit dividend, with remainder
  • 1-digit divisor, 6-digit dividend, no remainder
  • 1-digit divisor, 6-digit dividend, with remainder
  • 1-digit divisor, 7-digit dividend, no remainder
  • 1-digit divisor, 7-digit dividend, with remainder
  • 2-digit divisor, 5-digit dividend, no remainder
  • 2-digit divisor, 5-digit dividend, with remainder
  • 2-digit divisor, 6-digit dividend, no remainder
  • 2-digit divisor, 6-digit dividend, with remainder
  • 2-digit divisor, 7-digit dividend, no remainder
  • 2-digit divisor, 7-digit dividend, with remainder
  • 3-digit divisor, 6-digit dividend, no remainder
  • 3-digit divisor, 6-digit dividend, with remainder
  • 3-digit divisor, 7-digit dividend, no remainder
  • 3-digit divisor, 7-digit dividend, with remainder
    (0-2 decimal digits)
  • Divide a whole number or a decimal by a whole number, need to add zeros to the dividend , rounding the answers to three decimals

Convert measuring units using long division & multiplication

Primary Grade Challenge Math by Edward Zaccaro

A good book on problem solving with very varied word problems and strategies on how to solve problems. Includes chapters on: Sequences, Problem-solving, Money, Percents, Algebraic Thinking, Negative Numbers, Logic, Ratios, Probability, Measurements, Fractions, Division. Each chapter&rsquos questions are broken down into four levels: easy, somewhat challenging, challenging, and very challenging.

Exponents

Place value/Rounding

    (up to 9 digits) (up to 12 digits)
  • Write a number given in expanded form in normal form (up to 9 digits), the parts are scrambled
  • Write a number given in expanded form in normal form (up to 12 digits), the parts are scrambled (up to 6 decimal digits), the parts are scrambled
    - rounding to the underlined digit, up to rounding to the nearest million - round to the underlined digit, up to rounding to the nearest trillion

Algebra

Key to Algebra Workbooks

Key to Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language, and examples are easy to follow. Word problems relate algebra to familiar situations, helping students to understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system.

Fractions vs. Decimals

Decimal Addition and Subtraction

Key to Decimals Workbooks

This is a workbook series by Key Curriculum Press that begins with basic concepts and operations on decimals. Then the books cover real-world uses of decimals in pricing, sports, metrics, calculators, and science.

The set includes books 1-4.

Decimal Multiplication

Mental multiplication

Decimal Division

Measuring units

Customary system

Convert measuring units using long division & multiplication (paper & pencil) or mental math

Convert using a calculator, with decimals

  • Convert between inches, feet, and yards - use a calculator
  • Convert between miles, yards, and feet 1 - use a calculator
  • Convert between miles, yards, and feet 2 - use a calculator
  • Convert between tons, pounds, and ounces with decimals - use a calculator
  • Convert between various customary units with decimals - use a calculator
  • Convert between mm, cm and m - using decimals
  • Convert between mm, cm, m, and km - using decimals
  • Convert between ml & l and g & kg - using decimals
  • All metric units mentioned above - mixed practice - using decimals
  • Metric system: convert between the units of length (mm, cm, dm, m, dam, hm, km)
  • Metric system: convert between the units of weight (mg, cg, dg, g, dag, hg, kg)
  • Metric system: convert between the units of volume (ml, cl, dl, L, dal, hl, kl)
  • Metric system: convert between the units of length, weight, and volume

Ratio

Percent

Prime factorization, GCF, and LCM

Fraction addition and subtraction

Fraction multiplication

In all fraction multiplication and division problems, it helps to simplify before you multiply.

Fraction division

Convert fractions to mixed numbers and vv

Simplify or equivalent fractions

Fractions vs. Decimals

Integers

Coordinate grid

Addition & subtraction

Addition and subtraction of integers are beyond the Common Core Standards for grade 6 but some curricula or standards may include them in 6th grade.

Multiplication & Division

Multiplication and division of integers are beyond the Common Core Standards for grade 6 but the worksheets links are included here for completeness sake, as some curricula or standards may include them in 6th grade.

Geometry

Area- these worksheets are done in the coordinate grid.

Volume & surface area

Since these worksheets below contain images of variable sizes, please first check how the worksheet looks like in print preview before printing. If it doesn't fit, you can either print it scaled (such as at 90%), or make another one by refreshing the worksheet page (F5) until you get one that fits.

  • Find the volume of a rectangular prism with fractional edge lengths (easy: halves, thirds, and fourths the whole number part is max 1)
  • Find the volume of a rectangular prism with fractional edge lengths (easy: halves, thirds, and fourths the whole number part is max 2)
  • Find the volume of a rectangular prism with fractional edge lengths (challenge: fractions up till sixths)
  • Find the volume or surface area of rectangular prisms (easy)
  • Find the volume or surface area of rectangular prisms (using decimals)
  • Problem solving: find the volume/surface area/edge length of cube when surface area or volume is given

Formula Area of a Cylinder

This page examines the properties of a right circular cylinder. A cylinder has a radius (r) and a height (h) (see picture below).

This shape is similar to a can. The surface area is the area of the top and bottom circles (which are the same), and the area of the rectangle (label that wraps around the can).

The Cylinder Area Formula

The picture below illustrates how the formula for the area of a cylinder is simply the sum of the areas of the top and bottom circles plus the area of a rectangle. This rectangle is what the cylinder would look like if we 'unraveled' it.

Below is a picture of the general formula for area.

Practice Problems on Area of a Cylinder

Problem 1

What is the area of the cylinder with a radius of 2 and a height of 6?

Show Answer

Problem 2

What is the area of the cylinder with a radius of 3 and a height of 5?

Show Answer

Problem 3

What is the area of the cylinder with a radius of 6 and a height of 7?

Show Answer


List of Area Worksheets

The children in the 2nd grade and 3rd grade enhance practice with this interesting collection of pdf worksheets on finding the area by counting unit squares. Included here area exercises to count the squares in the irregular figures and rectangular shapes.

Give learning a head start with these finding the area of a square worksheets. Figure out the area of squares using the formula, determine the side lengths, find the length of the diagonals and calculate the perimeter using the area as well.

Strengthen skills in finding the area of a rectangle with these pdf worksheets featuring topics such as determine the area of rectangles, area of rectilinear shapes, rectangular paths and solve word problems. Recommended for grade 3, grade 4, grade 5 and up.

Augment your practice on finding the area with our area of rectilinear figures worksheets! With two or more non-overlapping rectangles composing them, these rectilinear shapes require adding the areas of those non-overlapping parts to arrive at their area.

Focusing on finding the area of triangles, this set of worksheets features triangles whose dimensions are given as integers, decimals and fractions involving conversion to specified units as well. Approx. grade levels: 5th grade, 6th grade and 7th grade.

The area of a parallelogram worksheets comprise adequate skills to find the area of a parallelogram, compute the value of the missing dimensions - base or height, practice finding the area by converting to specific units and more. The exercises are presented as geometric illustrations and also in word format.

This collection of area worksheets encompasses a variety of PDFs to find the area of trapezoids whose dimensions are given as integers, fractions and decimals. Determine the missing parameters by substituting the values in the formulas, solve exercises involving unit conversions too.

Emphasizing on how to find the area of a rhombus the worksheets here contain myriad PDFs to practice the same with dimensions presented as integers, decimals and fractions. Find the diagonal lengths, missing parameters, compute the area, learn to convert to a specified unit and much more.

Improve efficiency in finding the area of kites with these printable worksheets comprising illustrations and exercises in word format. Calculate the area of the kite, find the missing diagonal lengths using the area and much more!

Calculate the area of quadrilaterals whose dimensions are presented as whole numbers and fractions. The worksheets on area of a quadrilateral consist of exercises on rectangles, trapezoids, kites in the form of illustrations, on grids and in word format. Practice conversion to a specified unit in the process.

Reaffirm the concept of finding the area of a circle by using these practice worksheets. Learn to find the area or circumference using the given radius or diameter, calculate the area and circumference, compute the radius and diameter from the area or circumference given and a lot more.

An Annulus is the region enclosed between two concentric circles. How about calculating their area? Plug into this compact set of printable worksheets and practice calculating the area of those circular rings!

The children of grade 5, grade 6 and grade 7 can reinforce their skills in finding the area of mixed shapes by practicing this set of printable worksheets.

Incorporate these area of polygons worksheets comprising examples and adequate exercises to find the area of regular polygons like triangles, quadrilaterals and irregular polygons using the given side lengths, circumradius and apothem. Free worksheets are available for practice.

The area of compound shapes worksheets consist of a combination of two or more geometric shapes, find the area of the shaded parts by adding or subtracting the indicated areas, calculate the area of rectilinear shapes (irregular figures) and rectangular paths as well. This practice set is ideal for 4th grade through 7th grade.

Develop practice in finding the area of a segment of a circle with these practice pdfs. Adequate exercises in finding the area of the triangle and the area of the sector using one of the parameters given are sure to help students master calculating the area of the segment in no time.


Geometry of Vertical Image

In order to understand mission flight planning, you need to understand the geometry of the image as it is formed within the camera. The size of the CCD array and lens focal length coupled with flying altitude (above ground) determines the image scale or the ground resolution of the image. Therefore, it is essential to the work of the flight planner to have all of this information understood and available before starting to design a mission.

In photogrammetry, we usually deal with three types of imagery (photography), They are defined in term of the angle that the camera optical axis makes with the vertical (nadir), those are:

  1. true vertical photography: ±0º from nadir
  2. tilted or near-vertical photography > 0º but less than ±3º – Most used –
  3. oblique photography: between ±35º degree and ±55º off nadir

For the purpose of this course, we will focus only on the first two types, and that is vertical and near-vertical photography.

Figure 4.3 illustrates the basic geometry of a vertical photograph or image. By vertical photograph or image, we mean an image taken with a camera that is looking down at the ground. As the aircraft moves, so does the camera, and this makes it impossible to take a true vertical image. Therefore, vertical image definition allows a few degrees deviation from the nadir (the line connecting the lens frontal point and the point on the ground that is exactly beneath the aircraft). In summary, a vertical image is an image that is either looking straight down to the ground or is looking a few degrees to either side of the aircraft.

Scale of Vertical Image

As the sun's rays hit the ground, they reflect back toward the camera, and some actually enter the camera through the lens. This physical phenomenon enables us to express the ground-image relation using trigonometric principles. In Figure 4.3, ground point A is projected at image location a' and ground point B is projected at image location b' on the film. From such geometry, the film four corners a' b' c' d' cover an area on the ground represented by the square ABCD. Such relations not only enable us to compute the ground coverage of a photograph (image) but also enable us to compute the scale of such a photograph or image.

The scale of an image is the ratio of the distance on the image to the corresponding distance on the ground. In Figure 4.4, the distance on the ground AB will be projected on the image on line ab, therefore, the image scale can be computed using the following formula:

Equation 1: scale = distance ab distance AB

Analyzing the two triangles (the small triangle with base ab and the large triangle with base AB) of Figure 4.4, one can also conclude, using the similarity of triangles principle, that the scale is also equal to:

Equation 2: scale = lens focal length (f) Flying height (H)

Scale is expressed either in a unitless ratio such as 1/12,000 (or 1:12,000) or in pronounced units ratio such as 1 in. = 1,000 ft (or 1”=1,000’).

Examples on Scale Computations

The following two examples will walk you step by step through the process of computing scales for imagery produced from a film-based camera and from a digital camera. In digital cameras, the scale does not play any role in defining the image quality, as is the case with film-based camera. In digital cameras, we use the Ground Sampling Distance (GSD) to describe the resolution quality of the image while in film-based cameras we use the film scale.

Scale from Film Camera

Aerial photographs were acquired from an altitude of 6,000 ft AMT (Above Mean Terrain) with a film-based aerial camera with lens focal length of 6 inches. Determine the scale of the resulting photography.

From Figure 4.4 and equations 1 & 2,

Scale = lens focal length (f) Flying height (H) = distance ab distance AB

Scale = 6 in. 6 , 000 ft x 12 in/ft = distance ab distance AB

Scale from Digital Camera

Aerial imagery was acquired with a digital aerial camera with lens focal length of 100 mm and CCD size of 0.010 mm (or 10 microns). The resulting imagery had a ground resolution of 30 cm (1 ft). Determine the scale of the resulting imagery.

From Figure 4.4 and equation 1, assume that the distance ab represents the physical size of one pixel or CCD, which is 0.010 mm, and the distance AB is the ground coverage of the same pixel or 30 cm.

Scale = distance ab distance AB

Scale = 0.010 mm 30 cm x 10 mm/cm = 0.010 300 = 1 300 / 0.010 = 1 30 , 000

Practice Scale Computation Example:

Aerial imagery was acquired with a digital aerial camera with lens focal length of 50 mm and CCD size of 0.020 mm (or 20 microns). The resulting imagery had a ground resolution of 60 cm (2 ft). Determine the scale of the resulting imagery.

Scale = 0.020 mm 60 cm x 10 mm/cm = 0.020 600 = 1 30 , 000

Imagery Overlap

Imagery acquired for photogrammetric processing is flown with two types of overlap: Forward Lap and Side Lap. The following two subsections will describe each type of imagery overlap.

Forward Lap

Forward lap, which is also called end lap, is a term used in photogrammetry to describe the amount of image overlap intentionally introduced between successive photos along a flight line (see Figure 4.5). Flight 3 illustrates an aircraft equipped with a mapping aerial camera taking two overlapping photographs. The centers of the two photographs are separated in the air with a distance B. Distance B is also called air base. Each photograph of Figure 4.5 covers a distance on the ground equal to G. The overlapping coverage of the two photographs on the ground is what we call forward lap.

This type of overlap is used to form stereo-pairs for stereo viewing and processing. The forward lap is measured as a percentage of the total image coverage. Typical value for the forward lap for photogrammetric work is 60%. Because of the light weight of the UAS, we expect substantial air dynamic and therefore substantial rotations of the camera (i.e., crab) therefore, I recommend the amount of forward lap to be at least 70%.

Side Lap

Side lap is a term used in photogrammetry to describe the amount of overlap between images from adjacent flight lines (see Figure 4.6). Figure 4.6 illustrates an aircraft taking two overlapping photographs from two adjacent flight lines. The distance in the air between the two flight lines (W) is called lines spacing.

This type of overlap is needed to make sure that there are no gaps in the coverage. The side lap is measured as a percentage of the total image coverage. The typical value for the side lap for photogrammetric work is 30%. However, because of the light weight of the UAS, we expect substantial air dynamic and therefore substantial rotations of the camera (i.e. crab), and therefore I recommend using at least 40% side lap.

Image Ground Coverage

Ground coverage of an image is the area on the ground (the square ABCD of Figure 4.3) covered by the four corners of the photograph a'b'c'd' of Figure 4.3. Ground coverage of a photograph is determined by the camera internal geometry (focal length and the size of the CCD array) and the flying altitude above ground elevation.

Example on Image Ground Coverage:

A digital camera has an array size of 12,000 pixels by 6,000 pixels (Figure 4.7). If the physical CCD size is 0.010 mm (10 um) camera, how much area in acres will each image cover on the ground if the resulting ground resolution (GSD) of a pixel is 1 foot?

Ground coverage across the width (W) of the array = 12,000 pixels x 1 ft/pixel = 12,000 ft

Ground coverage across the height (L) of the array= 6,000 pixels x 1 ft/pixel = 6,000 ft

Covered area per image = W x L = 12 , 000 ft x 6 , 000 ft = 72 , 000 , 000 ft 2 = 72 , 000 , 000 43 , 560 = 1652.892 acres


News & Updates

Monday March 29th, 2021 Spelling Distance Learning Assignments Hey all,
So today something that has been in the works for a while and I think it's (finally) ready for the public.
Now on the Spelling Worksheet Maker you'll see an option for 'Distance Learning'.
Just click on that and instead of outputing worksheets it'll generate distance learning assignments.
And as a bonus, any previously created spelling list is compatible with the distance learning option. Huzzah!

I've been testing and tweaking it for a few months now, so I'm absolutely 100% positive there are no bugs or glitches.
That being said. when you find any bug or glitches please let me know in the comments. :P

Easter Coloring Sheets Also just in time for easter there are a few coloring sheets available here:
Coloring Sheets

Robert Smith (Admin)
[email protected]


Reducing the Scale Factor

The methods above to convert a measurement assume the scale factor is in the form of 1:n or 1/n, which means some additional work is needed if the ratio is 2:3, for example. When the scale factor is not in an even 1:n ratio, you’ll need to reduce it to 1:n.

Use our ratio calculator to reduce a ratio. You can also reduce a ratio by dividing both the numerator and the denominator by the numerator.

For example: reduce 2/3 by dividing both numbers by 2, which would be 1/1.5 or 1:1.5.


Watch the video: Enlargements (August 2022).