﻿ Area of circle, formula and illustrated lesson: how to calculate the area of a circle

# 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.

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

### Practice Problems

Remember the Formula:

$$Area = \pi \cdot r^2 \\ A = \pi \cdot ( 22')^2 \\ A = \pi \cdot 484 \\ A = \pi \cdot 1520.53084433746 \text{ square feet} \\ \\ Area = \boxed{ 1520.5 } \\ \text {Rounded to nearest tenth}$$

Remember the Formula:

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

Remember the Formula:

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

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

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

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?

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

Divide Diameter in half to calculate radius:

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

Use area formula

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

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 = $$3 \cdot 2 =6$$
A = $$\pi (3)^2$$ A = $$\pi (6)^2$$
A = $$9 \pi$$ A = $$36 \pi$$
$$A_{larger} = \color{red}{ 4} \cdot A_{smaller} \\ A_{larger} = \color{red}{4} \cdot 9\pi \\ A_{larger} = 36 \pi$$

This relationship holds true no matter what radius you pick

Let's make the original radius = 5 .

Smaller Circle Larger Circle
radius $$=5$$ radius $$=5 \cdot 2= 10$$
A = $$\pi (5)^2$$ A = $$\pi (10)^2$$
A = $$25 \pi$$ A = $$100 \pi$$
$$A_{larger} = \color{red}{ 4} \cdot A_{smaller} \\ A_{larger} = \color{red}{4} \cdot 25\pi \\ A_{larger} = 100 \pi$$

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