Lesson 3.4: Circles and Spheres

New Concept

This lesson introduces key geometric formulas. You'll learn to calculate the circumference and area of circles, and the volume of spheres, using the radius, diameter, and the constant $\pi$.

What’s next

Next, you'll apply these formulas through guided examples and practice cards to calculate area, circumference, and volume.

Property

The distance from the center of a circle to any point on the circle itself is called the radius of the circle.
The diameter of a circle is the length of a line segment joining two points on the circle and passing through the center. Thus, the diameter of a circle is twice its radius.
The perimeter of a circle is called its circumference.
The circumference $C$ of a circle is given by

where $d$ is the diameter of the circle. The Greek letter $\pi$ (pi) stands for an irrational number: $\pi = 3.141592654 ...$

Examples

• A circular pool has a diameter of 10 meters. Its circumference is $C = \pi \times 10 \approx 31.42$ meters.
• A bicycle wheel has a radius of 14 inches. Its diameter is $2 \times 14 = 28$ inches, so its circumference is $C = \pi \times 28 \approx 87.96$ inches.
• If a running track has a circumference of 400 meters, its diameter can be found by $d = \frac{C}{\pi} = \frac{400}{\pi} \approx 127.32$ meters.

Explanation

Circumference is the special name for a circle's perimeter. It's the distance around the circle's edge. This distance is always a little more than 3 times the circle's diameter, a constant ratio we call pi ($\pi$).

Property

The area enclosed by a circle is given by

where $r$ is the radius of the circle. An exponent tells us how many times the base occurs as a factor in a product. So $r^2$ means $r \times r$. We compute powers before we compute the rest of a product.

Examples

• A pizza has a radius of 7 inches. Its area is $A = \pi \times 7^2 = 49\pi \approx 153.94$ square inches.
• A circular garden has a diameter of 20 feet. Its radius is 10 feet, so its area is $A = \pi \times 10^2 = 100\pi \approx 314.16$ square feet.
• A circular rug has an area of 50 square feet. Its radius is found by solving $r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{50}{\pi}} \approx \sqrt{15.91} \approx 3.99$ feet.

Explanation

The area of a circle tells you the amount of space inside it. To find it, you first square the radius (multiply it by itself), and then multiply that result by pi ($\pi$). This gives the total surface coverage.

Property

The volume of a sphere is given by

where $r$ is the radius of the sphere. Recall that $r^3$, which we read as '$r$ cubed,' means $r \times r \times r$.

Examples

• A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters.
• A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters.
• A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$, which simplifies to $27 = r^3$, so the radius is $r = \sqrt[3]{27} = 3$ inches.

Explanation

Volume measures the space inside a 3D shape, like a ball or a planet. For a sphere, you cube the radius (multiply it by itself three times), then multiply by pi ($\pi$), and finally multiply by the fraction $\frac{4}{3}$.

Property

The diameter of a circle is twice its radius.

Circumference of a circle

Area of a circle

Volume of a sphere

Examples

• For a circle with a radius of 5 cm, the circumference is $C = 2\pi(5) = 10\pi \approx 31.42$ cm and the area is $A = \pi(5)^2 = 25\pi \approx 78.54$ square cm.
• A sphere has a diameter of 12 inches, so its radius is 6 inches. Its volume is $V = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi \approx 904.78$ cubic inches.
• A circular plate has a circumference of 30 inches. Its diameter is $d = \frac{30}{\pi} \approx 9.55$ inches, and its radius is approximately $4.775$ inches.

Explanation

This card is a quick reference for all the key formulas for circles and spheres. Notice how the radius ($r$) or diameter ($D$) is the key measurement you need to find circumference, area, or volume.