Area of a Circle and its formula

Practice Problems & examples

Formula for Area of circle

The formula to find a circle's area $$ \pi $$ (radius)2 usually expressed as $$ \pi \cdot r^2 $$ where r is the radius of a circle.

area of circle demo

The area of a circle is all the space inside a circle's circumference.
In the picture on the left, the area of the circle is the part of the circle that is changing color.

Explore and discover the relationship bet wen the area formula , the radius of a circle and its graph with with our interactive applet

Practice Problems

Problem 1

What is the area of the circle on the left?
Round your answer to the nearest tenth.

Circle's Area

Remember the Formula:

$$ Area = \pi \cdot r^2 \\ A = \pi \cdot ( 22')^2 \\ A = \pi \cdot 1,520 \text{ square feet} \\ A = 4775.220833456486\text{ square feet} \\ Area = \boxed{ 4775.2 } $$

Problem 2

What is this circle's area?
Round your answer to the nearest tenth.

Area of Circle Two

Remember the Formula:

$$ Area = \pi \cdot r^2 \\ A = \pi \cdot ( 5")^2 \\ A = \pi \cdot 25 \text{ square inches} \\ A = 78.53981633974483 \text{square inches} \\ A = 78.5 \text{ square inches, rounded to nearest tenth} $$

Problem 3

What is the area of a circle with a radius of 7 centimeters?
Round your answer to the nearest hundredth

Circle's Area

Remember the Formula:

$$ Area = \pi \cdot r^2 \\ A = \pi \cdot (7 \text{ centimeters} )^2 \\ A = 153.93804002589985 \text{ square centimeters} \\ \boxed { A = 153.94 \text{ square centimeters, rounded to nearest hundredth}} \\ $$

Problem 4

What is the radius of a circle if its area is 120 in2? (Round your answer to the nearest hundredth of an inch)

Circle's Area

Use the area formula ... but this time solve the radius

$$ A = \pi r^2 \\ 120 = \pi r^2 \\ \frac{120}{\pi} = r^2 \\ 38.197 = r^2 \\ \sqrt{38.197} = r \\ \boxed {r=6.18 \text{ inches }} $$

Problem 5

What is the diameter of a circle if its area is 360 in2? (Round your answer to the nearest hundredth of an inch)

Circle's Area

Like the last problem, we are given area and need to solve for radius ; However, this time, we need to then do one more step - find the diameter

$$ A = \pi r^2 \\ 360 in^2= \pi r^2 \\ \frac{360}{\pi} = r^2 \\ 114.59155902616465= r^2 \\ \sqrt{114.59155902616465} = r \\ r=10.704744696916627 \text{ inches } \\ $$

Now, that we have found the radius, how do we find the diameter?

$$ diameter = 2 \cdot radius \\ = 2 \cdot 10.704744696916627 \\ =21.409489393833255 \\ \boxed{diameter =21.41 \text{ inches, rounded to nearest hundredth}} $$

Challenge Problems
challenge problems

A circle has a diameter of 12 inches. What is its area in terms of $$ \pi $$(Need a hint)

Divide Diameter in half to calculate radius:

Radius = $$ \frac{diameter}{2} = \frac{12}{2}= 6 $$

Use area formula

$$ A= \pi \cdot r^2 \\ A = \pi \cdot 6^2 \\ A = 36 \pi $$

Problem 7

If a circle's radius is doubled, then how much did its area increase?

Circle's Area

Since the formula for the area of a circle squares the radius, the area of the larger circle is always 4 (or 22) times the smaller circle. Think about it: You are doubling a number (which means ×2) and then squaring this (ie squaring 2) --which leads to a new area that is four times the smaller one.

You can see this relationship is true if you pick some actual values for the radius of a circle.

For instance, let's make the original radius = 3.
Smaller Circle Larger Circle
radius =3 radius =3*2=6
A = Π(3)2 A = Π(6)2
A = 9 Π A = 36Π
A = 9 Π × 4 = 36 Π

This relationship holds true no matter what radius you pick

Let's make the original radius = 5.

Smaller Circle Larger Circle
radius =5 radius =5×2= 10
A = Π(5)2 A = Π(10)2
A = 25Π A = 100Π
A = 25Π × 4 = 100 Π
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