# 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.

Diagram 1

Area of Circle Concept

The area of a circle is all the space inside a circle's circumference. In diagram 1, the area of the circle is indicated by the blue color.

The area is not actually part of the circle. Remember a circle is just a locus of points. The area is enclosed inside that locus of points.

### Practice Problems

##### Problem 1

Remember the Formula:

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

##### Problem 2

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

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

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

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

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

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 = $$100 \pi$$

$$A_{larger} = \color{red}{4} \cdot A_{smaller} \\ A_{larger} = \color{red}{4} \cdot 25\pi \\ A_{larger} = 100 \pi$$