Circles room a common shape. You view them all over—wheels ~ above a car, Frisbees passing with the air, compact discs carrying data. These are all circles.

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A circle is a two-dimensional figure similar to polygons and quadrilaterals. However, circles space measured differently than these other shapes—you even have to usage some different terms to describe them. Let’s take it a look at this interesting shape.


A circle represents a set of points, every one of which space the exact same distance away from a fixed, center point. This fixed suggest is dubbed the center. The distance from the center of the circle to any allude on the circle is dubbed the radius.

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When 2 radii (the many of radius) are put together to form a line segment across the circle, you have a diameter. The diameter the a one passes with the center of the circle and also has that endpoints on the one itself.

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The diameter of any circle is 2 times the length of the circle’s radius. It have the right to be represented by the expression 2r, or “two time the radius.” for this reason if you understand a circle’s radius, you deserve to multiply that by 2 to discover the diameter; this also way that if you know a circle’s diameter, you can divide by 2 to find the radius.


Example

Problem

Find the diameter the the circle.

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d = 2r

d = 2(7)

d = 14

The diameter is two times the radius, or 2r. The radius that this one is 7 inches, therefore the diameter is 2(7) = 14 inches.

Answer

The diameter is 14 inches.


Example

Problem

Find the radius the the circle.

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The radius is fifty percent the diameter, or

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. The diameter that this circle is 36 feet, so the radius is
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 feet.

Answer

The radius is 18 feet.


Circumference


The distance around a one is dubbed the The distance about a circle, calculation by the formula C =

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d.


")">circumference
. (Recall, the distance about a polygon is the perimeter.)

One interesting property about circles is the the ratio of a circle’s circumference and also its diameter is the same for all circles. No matter the dimension of the circle, the ratio of the circumference and diameter will certainly be the same.

Some actual dimensions of various items are noted below. The measurements are exact to the nearest millimeter or quarter inch (depending top top the unit of measure up used). Look at the ratio of the circumference to the diameter because that each one—although the items room different, the proportion for every is around the same.

 


Item

Circumference (C) (rounded come nearest hundredth)

Diameter (d)

Ratio

Cup

253 mm

79 mm

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Quarter

84 mm

27 mm

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Bowl

37.25 in

11.75 in

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The circumference and also the diameter space approximate measurements, due to the fact that there is no precise method to measure up these dimensions exactly. If you were able to measure up them much more precisely, however, girlfriend would uncover that the proportion  would move towards 3.14 for each that the items given. The mathematical surname for the ratio  is The proportion of a circle’s circumference come its diameter. Pi is denoted through the Greek letter

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. It is frequently approximated together 3.14 or
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.


")">pi
, and also is stood for by the Greek letter .

 is a non-terminating, non-repeating decimal, so the is impossible to write it the end completely. The first 10 number of  are 3.141592653; it is frequently rounded to 3.14 or approximated as the portion . Keep in mind that both 3.14 and also  are approximations of, and also are provided in calculations whereby it is not necessary to be precise.

Since you understand that the ratio of circumference to diameter (or ) is constant for all circles, you have the right to use this number to uncover the one of a one if you know its diameter.

 = , for this reason C = d

Also, since d = 2r, climate C = d = (2r) = 2r.

Circumference of a Circle

To uncover the circumference (C) of a circle, use one of the complying with formulas:

If you recognize the diameter (d) the a circle:

If you understand the radius (r) the a circle:


Example

Problem

Find the circumference of the circle.

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To calculate the circumference offered a diameter the 9 inches, use the formula . Usage 3.14 as an approximation for .

Since you space using one approximation for , you can not give specific measurement of the circumference. Instead, you use the price

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 to indicate “approximately equal to.”

Answer

The circumference is 9 or approximately 28.26 inches.


Example

Problem

Find the circumference of a circle v a radius of 2.5 yards.

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To calculate the circumference of a circle given a radius of 2.5 yards, usage the formula . Usage 3.14 as an approximation for.

Answer

The one is 5 or about 15.7 yards.


A circle has a radius that 8 inches. What is that circumference, rounded to the nearest inch?

A) 25 inches

B) 50 inches

C) 64 inches2

D) 201 inches


Show/Hide Answer

A) 25 inches

Incorrect. You multiplied the radius time ; the correct formula because that circumference as soon as the radius is given is The correct answer is 50 inches.

B) 50 inches

Correct. If the radius is 8 inches, the exactly formula because that circumference once the radius is provided is The correct answer is 50 inches.

C) 64 inches2

Incorrect. Girlfriend squared 8 inch to come at the price 64 inches2; this will provide you the area that a square with sides the 8 inches. Remember that the formula because that circumference when the radius is offered is . The exactly answer is 50 inches.

D) 201 inches

Incorrect. That looks choose you squared 8 and also then multiplied 64 through  to come at this answer. Remember the the formula for circumference once the radius is offered is . The correct answer is 50 inches.

Area


 is critical number in geometry. You have already used the to calculate the one of a circle. You use  when you are figuring the end the area that a circle, too.

Area of a Circle

To discover the area (A) the a circle, use the formula:


Example

Problem

Find the area of the circle.

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To find the area that this circle, use the formula .

Remember to compose the price in terms of square units, since you space finding the area.

Answer

The area is 9 or about 28.26 feet2.


A button has a diameter of 20 millimeters. What is the area of the button? use 3.14 together an approximation the .

A) 62.8 mm

B) 314 mm2

C) 400 mm2

D) 1256 mm2


Show/Hide Answer

A) 62.8 mm

Incorrect. You discovered the circumference of the button: 20 • 3.14 = 62.8. To uncover the area, usage the formula . The correct answer is 314 mm2.

B) 314 mm2

Correct. The diameter is 20 mm, so the radius must be 10 mm. Then, making use of the formula , you find

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 mm2.

C) 400 mm2

Incorrect. Friend squared 20 to arrive at 400 mm2; this provides you the area of a square v sides of length 20, no the area the a circle. To find the area, usage the formula . The correct answer is 314 mm2.

D) 1256 mm2

Incorrect. It looks prefer you squared 20 and then multiplied by . 20 is the diameter, no the radius! To discover the area, use the formula . The correct answer is 314 mm2.

Composite Figures


Now that you know exactly how to calculate the circumference and also area that a circle, you deserve to use this understanding to find the perimeter and also area that composite figures. The trick to figuring the end these varieties of difficulties is to recognize shapes (and parts of shapes) in ~ the composite figure, calculate your individual dimensions, and then include them together.

For example, look at the image below. Is it feasible to uncover the perimeter?

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The first step is come identify easier figures within this composite figure. You can break it down right into a rectangle and a semicircle, as shown below.

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You know just how to find the perimeter of a rectangle, and also you know exactly how to find the one of a circle. Here, the perimeter the the 3 solid sides of the rectangle is 8 + 20 + 20 = 48 feet. (Note that only three political parties of the rectangle will add into the perimeter of the composite figure due to the fact that the other side is not at one edge; that is spanned by the semicircle!)

To find the circumference of the semicircle, usage the formula  with a diameter that 8 feet, then take fifty percent of the result. The circumference of the semicircle is

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, or around 12.56 feet, so the total perimeter is around 60.56 feet.

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Example

Problem

Find the perimeter (to the nearest hundredth) the the composite figure, consisted of of a semi-circle and also a triangle.

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Identify smaller shapes within the composite figure. This figure contains a semicircle and a triangle.

Diameter (d) = 1

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Circumference that semicircle =

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 or about 1.57 inches

Find the one of the circle. Then division by 2 to discover the one of the semi-circle.

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 inches

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Find the full perimeter by including the circumference of the semicircle and also the lengths that the two legs. Since our measure of the semi-circle’s one is approximate, the perimeter will be one approximation also.

Answer

Approximately 3.57 inches


Example

Problem

Find the area that the composite figure, consisted of of three-quarters that a circle and also a square, to the nearest hundredth.

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Identify smaller shapes within the composite figure. This figure consists of a circular region and a square. If you discover the area that each, girlfriend can find the area of the entire figure.

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Find the area that the square.

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*

*

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.

Find the area of the one region. The radius is 2 feet.

Note the the region is  of a entirety circle, therefore you should multiply the area the the one by . Use 3.14 as an approximation because that .

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4 feet2 +

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 feet2 = around 13.42 feet2

Add the two areas together. Because your measure up of the circular’s area is approximate, the area of the figure will be an approximation also.

Answer

The area is around 13.42 feet2.


What is the area (to the nearest hundredth) of the figure shown below? (Both rounded areas are semi-circles.)

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A) 16.56 in2

B) 7.14 in2

C) 4 in2

D) 3.14 in2


Show/Hide Answer

A) 16.56 in2

Incorrect. The looks choose you calculate the area that a circle using a radius that 2; in this figure, the radius of each circle is 1. To discover the area that the figure, imagine the two semi-circles are placed together to create one circle. Then calculate the area that the circle and include it come the area the the square. The correct answer is 7.14 in2.

B) 7.14 in2

Correct. Imagine the two semi-circles being put together to create one circle. The radius the the one is 1 inch; this means the area the the circle is

*
. The area the the square is 2 • 2 = 4. Adding those together yields 7.14 in2.

C) 4 in2

Incorrect. It looks favor you calculation the area that the square, however not the circle. Imagine the 2 semi-circles are placed together to develop one circle. Then calculation the area of the one and include it come the area the the square. The exactly answer is 7.14 in2.

D) 3.14 in2

Incorrect. It looks like you calculate the area the the circle, but not the square. Calculation the area that the square and add it come the area the the circle. The exactly answer is 7.14 in2.

See more: Convert 220 Mm Equals How Many Inches Converter, Millimeters To Inches Converter


Summary


Circles are crucial geometric shape. The distance roughly a one is dubbed the circumference, and the interior space of a one is dubbed the area. Calculating the circumference and also area that a circle requires a number called pi (), i m sorry is a non-terminating, non-repeating decimal. Pi is often approximated by the values 3.14 and . You can discover the perimeter or area of composite shapes—including shapes that save on computer circular sections—by using the circumference and also area formulas where appropriate.