How to Calculate Area of a Circle

X

This article was co-written by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math teacher at City University of San Francisco and previously worked in the math department of Saint Louis University. She has taught math at the elementary, middle, high, and college levels. She holds a master’s degree in education from Saint Louis University, majoring in management and supervision in education.

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One of the common problems in geometry is calculating the area of a circle based on known information. The formula for calculating the area of a circle is: $A$ $=$ $π$ $r$ $2$ ${displaystyle A=pi r^{2}}$ $How to Calculate Area of a Circle$ . This formula is quite simple, you just need to know the value of the radius to calculate the area of the circle. However, you also need to practice converting a given number of data units into terms that can be applied to this formula.

Table of Contents

Use radius to find area

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Determine the radius of the circle. The radius is the length from the center to the edge of the circle. Whichever direction you measure in, the radius is the same. The radius is also half the diameter of the circle. The diameter is the line segment that passes through the center and joins the two opposite sides of the circle. ^{[1] X Research Source}

The problem is usually given a radius. It is a bit difficult to determine the exact center of the circle, unless it is already given on the drawing provided by the problem.
In this example, let’s say the problem gives you a circle radius of 6 cm.

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Radius squared. The formula for the area of a circle is

A

=

π

r

2

{displaystyle A=pi r^{2}}

How to Calculate Area of a Circle

, where the variable

r

{displaystyle r}

r

represents the radius. This variable is squared up. ^{[2] X Research Source}

Don’t get confused and square the whole expression.
For example, a circle has a radius, $r$ $=$ $6$ ${displaystyle r=6}$ $r=6$ , We have $r$ $2$ $=$ $36$ ${displaystyle r^{2}=36}$ $r^{2}=36$ .

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Multiply by pi. Pi is a mathematical constant that represents the ratio between the circumference and diameter of a circle. It is denoted by the Greek letter

π

{displaystyle pi }

pi

. ^{[3] X Research Source} After rounding to decimals,

π

{displaystyle pi }

pi

close to 3.14. The correct decimal value actually goes on indefinitely. Normally, to represent the area of a circle accurately, we would write the answer in the notation

π

{displaystyle pi }

pi

. ^{[4] X Research Sources}

With the example of a circle with a radius of 6 cm, the area would be calculated as follows:
- $A$ $=$ $π$ $r$ $2$ ${displaystyle A=pi r^{2}}$ $How to Calculate Area of a Circle$
- $A$ $=$ $π$ $6$ $2$ ${displaystyle A=pi 6^{2}}$ $A=pi 6^{2}$
- $A$ $=$ $36$ $π$ ${displaystyle A=36pi }$ $A=36pi$ nice $A$ $=$ $36$ $($ $3$ $,$ $14$ $)$ $=$ $113$ $,$ $04$ ${displaystyle A=36(3,14)=113.04}$ $A=36(3.14)=113.04$

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Present the answer. Remember that when calculating area, units should always be presented with a “squared” sign (pronounced square). If the radius is in centimeters, the area will be square centimeters. If the radius is in meters, the area will be square meters. You also need to know how the question asks us to present the answer: in the notation

π

{displaystyle pi }

pi

or calculate the rounded decimal? If you don’t know, state it both ways. ^{[5] X Research Sources}

For a circle with a radius of 6 cm, the area will be 36 $π$ ${displaystyle pi }$ $pi$ cm ² or 113.04 cm ² .

Calculate area by diameter

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Measure or rewrite the diameter. In some problems or situations, you won’t know the radius. Instead, you will only know the length of the diameter of the circle. If the diameter is plotted in the problem diagram, you can use a ruler to measure it. Or, the problem will give the length of the diameter.

Suppose, you have a circle with a diameter of 20 cm.

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Divide the diameter in half. Remember that the diameter is twice as long as the radius. So no matter what diameter the problem gives you, just divide it in half and you will get the radius.

According to the example above, a circle with a diameter of 20 cm would have a radius of 20/2 = 10 cm.

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Apply the basic area stick formula. After converting diameter to radius, it’s time to use the formula

A

=

π

r

2

{displaystyle A=pi r^{2}}

How to Calculate Area of a Circle

to calculate the area of a circle. Assign the value of the radius and proceed with the remaining calculation as follows:

$A$ $=$ $π$ $r$ $2$ ${displaystyle A=pi r^{2}}$ $How to Calculate Area of a Circle$
$A$ $=$ $π$ $ten$ $2$ ${displaystyle A=pi 10^{2}}$ $A=pi 10^{2}$
$A$ $=$ $100$ $π$ ${displaystyle A=100pi }$ $A=100pi$

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Show the value of area. As a reminder, the unit of area of a circle will be accompanied by a “squared” sign. In this example, the diameter is in cm, so the radius is also in cm. So, the area will be in square centimeters. The answer here would be

100

π

{displaystyle 100pi }

100pi

cm ² .

You can also provide a decimal by substituting 3.14 for $π$ ${displaystyle pi }$ $pi$ . The result of the expression is (100)(3.14) = 314 cm ² .

Use perimeter to calculate area

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Learn about transformation formulas. If you know the circumference of a circle, you can use the transform formula to find the area of the circle. This transformation formula directly assigns the perimeter to calculate the area, you don’t need to find the radius. The new formula is:

$A$ $=$ $OLD$ $2$ $4$ $π$ ${displaystyle A={frac {C^{2}}{4pi }}}$ $A={frac {C^{2}}{4pi }}$

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Measure or write down the circumference. In some real world situations, you may not be able to measure the diameter or radius accurately. It is difficult to estimate the center of a circle if the diameter or center of the circle is not specified. For some round objects – such as a pizza pan or frying pan – you can use a tape measure to measure the circumference, which is much more accurate than measuring the diameter. ^{[6] X Research Source}

In this example, let’s say you have a circle (or a circular object) with a circumference of 42 cm.

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Use the circumference and radius relationship to transform the formula. The circumference of a circle is pi times the diameter or

OLD

=

π

d

{displaystyle C=pi d}

C=pi d

. Next, recall that the diameter is twice the radius, or

d

=

2

r

{displaystyle d=2r}

d=2r

. You can combine these two expressions to create the following relationship:

OLD

=

π

2

r

{displaystyle C=pi 2r}

C=pi 2r

. Rearrange the expression to isolate the variable r

r

{displaystyle r}

r

, we have: ^{[7] X Research source}

$OLD$ $=$ $π$ $2$ $r$ ${displaystyle C=pi 2r}$ $C=pi 2r$
$OLD$ $2$ $π$ $=$ $r$ ${displaystyle {frac {C}{2pi }}=r}$ ${frac {C}{2pi }}=r$ ….. (divide both sides by 2 $π$ ${displaystyle pi }$ $pi$ )

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Substitute the formula for calculating the area of a circle. Taking advantage of the relationship between circumference and radius, you will create a modified version of the formula for the area of a circle. Substituting the last expression into the original area formula, we have: ^{[8] X Research Source}

$A$ $=$ $π$ $r$ $2$ ${displaystyle A=pi r^{2}}$ $How to Calculate Area of a Circle$ ….. (the formula for the original area)
$A$ $=$ $π$ $($ $OLD$ $2$ $π$ $)$ $2$ ${displaystyle A=pi ({frac {C}{2pi }})^{2}}$ $A=pi ({frac {C}{2pi }})^{2}$ ….. (substitute the expression for r)
$A$ $=$ $π$ $($ $OLD$ $2$ $4$ $π$ $2$ $)$ ${displaystyle A=pi ({frac {C^{2}}{4pi ^{2}}})}$ $A=pi ({frac {C^{2}}{4pi ^{2}}})$ …..(squared fraction)
$A$ $=$ $OLD$ $2$ $4$ $π$ ${displaystyle A={frac {C^{2}}{4pi }}}$ $A={frac {C^{2}}{4pi }}$ …..(simple $π$ ${displaystyle pi }$ $pi$ in numerator and denominator)

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Apply the transform formula to calculate the area. Apply the rewritten transform formula with circumference instead of radius along with the information you have to find the exact area. Assign the value of the perimeter and perform the following calculation: ^{[9] X Research source}

In this example, you have perimeter $OLD$ $=$ $42$ ${displaystyle C=42}$ $C=42$ cm.
$A$ $=$ $OLD$ $2$ $4$ $π$ ${displaystyle A={frac {C^{2}}{4pi }}}$ $A={frac {C^{2}}{4pi }}$
$A$ $=$ $42$ $2$ $4$ $π$ ${displaystyle A={frac {42^{2}}{4pi }}}$ $A={frac {42^{2}}{4pi }}$ ….. (substitute value)
$A$ $=$ $1764$ $4$ $π$ ${displaystyle A={frac {1764}{4pi }}}$ $A={frac {1764}{4pi }}$ ….(calculated 42 ² )
$A$ $=$ $441$ $π$ ${displaystyle A={frac {441}{pi }}}$ $A={frac {441}{pi }}$ …..(divided by 4)

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Give the answer. Unless the circumference you have is a multiple of

π

{displaystyle pi }

pi

, otherwise your result will be a fraction with

π

{displaystyle pi }

pi

is the denominator. This answer is not wrong. You should present the answer for area in this way, or calculate the approximate answer by replacing pi with 3.14. ^{[10] X Research Source}

In this example, a circle with a circumference of 42 cm would have an area of $441$ $π$ ${displaystyle {frac {441}{pi }}}$ ${frac {441}{pi }}$ cm ²
If we want to calculate decimals, we have $441$ $π$ $=$ $441$ $3$ $,$ $14$ $=$ $140$ $,$ $4$ ${displaystyle {frac {441}{pi }}={frac {441}{3,14}}=140.4}$ ${frac {441}{pi }}={frac {441}{3,14}}=140.4$ . The area is approximately 140 cm ² .

Calculate the area in the shape of a fan

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Identify known or given information. Some problems will give you information about the shape of a circle, and they will ask you to calculate the total area of the circle. Read the passage carefully and look for information similar to, “A fan of circle O has an area of 15

π

{displaystyle pi }

pi

cm ² . Calculate the area of the circle O.” ^{[11] X Research Source}

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Determine the given fan shape. The fan shape of the circle is a division of the circle. A fan is defined by drawing two radii lines from the center to the edge of the circle. The space between those two radii is the fan shape. ^{[12] X Research Source}

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Calculate the angle at the center of the fan. Use a protractor to measure the angle between the two radii. Place the bottom edge of the protractor along a radius line, the center of the ruler coincident with the center of the circle. Then read the angle measure located at the position of the second radius forming the fan shape. ^{[13] X Research Source}

Make sure you measure the correct small angle between the two radii and not the larger outside angle. Usually, the problem you’re solving will give you this metric. The sum of the minor and major angles will be 360 degrees.
In some problems, the problem will give you the measure of the angle. For example: “The angle at the center of the fan is 45 degrees”, if you do not have the data, you will have to take the measurement.

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Apply the transform formula to calculate the area. Once you know the area of the fan and the measure of the angle at its center, you can apply the transform formula to find the area of the circle: ^{[14] X Research Source}

Acir=ASec360OLD{displaystyle A_{cir}=A_{sec}{frac {360}{C}}} $A_{{cir}}=A_{{sec}}{frac {360}{C}}$
- $A$ $c$ $i$ $r$ ${displaystyle A_{cir}}$ $A_{{cir}}$ is the total area of the circle
- $A$ $S$ $e$ $c$ ${displaystyle A_{sec}}$ $A_{{sec}}$ is the area of the fan shape
- $OLD$ ${displaystyle C}$ $OLD$ is the measure of the angle at the center

Steps

Use radius to find area

Calculate area by diameter

Use perimeter to calculate area

Calculate the area in the shape of a fan

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