- What Is The Area Of A Circle?
- How To Find The Area Of A Circle
- How to Calculate the Area of a Circle
- Area Of A Circle Using Circumference

A circle is not a square, but a circle's area (the amount of interior space enclosed by the circle) is measured in square units. Finding the area of a square is easy: length times width.

A circle, though, has only a **diameter**, or distance across. It has no clearly visible length and width, since a circle (by definition) is the set of all points equidistant from a given point at the center.

Yet, with just the diameter, or half the diameter (the **radius**), or even only the **circumference** (the distance around), you can calculate the area of any circle.

Recall that the relationship between the circumference of a circle and its diameter is always the same ratio, $3.14159265$, **pi**, or $\pi $. That number, $\pi $, times the square of the circle's radius gives you the area of the inside of the circle, in square units.

If you know the radius, $r$, in whatever measurement units (mm, cm, m, inches, feet, and so on), use the formula **π r ^{2}** to find area, $A$:

$A=\pi {r}^{2}$

The answer will be square units of the linear units, such as $m{m}^{2}$, $c{m}^{2}$, ${m}^{2}$, square inches, square feet, and so on.

Here is a circle with a radius of 7 meters. What is its area?

*[insert drawing of 14-m-wide circle, with radius labeled 7 m]*

$A=\pi \xb7{r}^{2}$

$A=\pi \times {7}^{2}$

$A=\pi \times 49$

$\mathbf{A}\mathbf{=}\mathbf{153.9380}{\mathbf{m}}^{\mathbf{2}}$

If you know the diameter, $d$, in whatever measurement units, take half the diameter to get the radius, $r$, in the same units.

Here is the real estate development of Sun City, Arizona, a circular town with a diameter of $1.07$ kilometers. What is the area of Sun City?

First, find half the diameter, given, to get the radius:

$\frac{1.07}{2}=0.535km=\mathbf{535}\mathbf{m}$

Plug in the radius into our formula:

$A=\pi \xb7{r}^{2}$

$A=\pi \times {535}^{2}$

$A=\pi \times \mathrm{286,225}$

$A=\mathrm{899,202.3572}{m}^{2}$

To convert square meters, ${m}^{2}$, to square kilometers, $k{m}^{2}$, divide by $\mathrm{1,000,000}$:

$\mathbf{A}\mathbf{=}\mathbf{0.8992}{\mathbf{km}}^{\mathbf{2}}$

Sun City's westernmost circular housing development has an area of nearly 1 square kilometer!

Try these area calculations for four different circles. Be careful; some give the radius, $r$, and some give the diameter, $d$.

Remember to take half the diameter to find the radius before squaring the radius and multiplying by $\pi $.

- A 406-mm bicycle wheel
- The London Eye Ferris wheel with a radius of 60 meters
- A 26-inch bicycle wheel
- The world's largest pizza had a radius of 61 feet, 4 inches (736 inches)

Do not peek at the answers until you do your calculations!

**A 406-mm bicycle wheel has a radius, $r$, of 203 mm:****The London Eye Ferris wheel's 60-meter radius:****A 26-inch bicycle wheel has a radius, $r$, of 13 inches:****The world's largest pizza with its 736-inch radius:**

$A=\pi {r}^{2}$

$A=\pi \times 203m{m}^{2}$

$A=637.7433m{m}^{2}$

$A=\pi {r}^{2}$

$A=\pi \times 60{m}^{2}$

$A=188.4955{m}^{2}$

$A=\pi {r}^{2}$

$A=\pi \times 13i{n}^{2}$

$A=530.9291i{n}^{2}$

$A=\pi {r}^{2}$

$A=\pi \times 736i{n}^{2}$

$A=\mathrm{1,701,788.17}i{n}^{2}$

That is $\mathrm{11,817.97}f{t}^{2}$ of pizza! Yum! Anyway, how did you do on the four problems?

If you have no idea what the radius or diameter is, but you know the circumference of the circle, $C$, you can *still* find the area.

Circumference (the distance around the circle) is found with this formula:

$C=2\pi r$

That means we can take the circumference formula and "solve for $r$," which gives us:

$r=\frac{C}{2\pi}$

We can replace $r$ in our original formula with that new expression:

$A=\pi {\left(\frac{C}{2\pi}\right)}^{2}$

That expression simplifies to this:

$A=\frac{{C}^{2}}{4\pi}$

That formula works every time!

Here is a beautiful, *reasonable-sized* pizza you and three friends can share. You happen to know the circumference of your pizza is $50.2655$ inches, but you do not know its total area. You want to know how many square inches of pizza you will each enjoy.

*[insert cartoon drawing of typical 16-inch pizza but do not label diameter]*

Substitute $50.2655$ inches for $C$ in the formula:

$A=\frac{{50.2655}^{2}}{4\pi}$

$A=\frac{\mathrm{2,526.6204}}{4\pi}$

$\mathbf{A}\mathbf{=}\mathbf{201.0620}{\mathbf{in}}^{\mathbf{2}}$

Equally divide that total area for a full-sized pizza among four friends, and you each get **$50.2655i{n}^{2}$** of pizza! That's about a third of a square foot for each of you! Yum, yum!

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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