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>Spheres>How to Find Volume and Surface Area of a Sphere

How to Find Volume and Surface Area of a Sphere

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Now it’s time to learn how to find the volume and the surface area of spheres.

Surface Area of a Sphere

Imagine an inflated beach ball. It consists of a thin plastic cover filled with air. The surface area of the ball is then equal to the area of the plastic cover of the beach ball. Here you’ll learn to find the surface area of a sphere.

Formula

Surface Area of a Sphere

A = 4 \cdot π \cdot r^{2}

where $π \approx 3.14$ and $r$ is the radius of the sphere.

Surface area of sphere

Example 1

What is the surface of a sphere with a radius of $2.5 cm$ ?

\begin{array}{l} A & = 4 \cdot π \cdot 2 . 5^{2} \approx 4 \cdot 3.14 \cdot 6.25 \\ \approx 78.5 {cm}^{2} \end{array}

A = 4 \cdot π \cdot 2 . 5^{2} \approx 4 \cdot 3.14 \cdot 6.25 = 78.5 {cm}^{2}

Example 2

What is the surface area of a sphere with a diameter of $8 cm$ ?

The first thing you need is the radius. It’s half the diameter, which makes it

r = \frac{8 cm}{2} = 4 cm

Thus, the surface area is

\begin{array}{l} A & = 4 \cdot π \cdot 4^{2} \approx 4 \cdot 3.14 \cdot 16 \\ \approx 200.96 {cm}^{2} \end{array}

A = 4 \cdot π \cdot 4^{2} \approx 4 \cdot 3.14 \cdot 16 = 200.96 {cm}^{2}

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Volume of a Sphere

Formula

Volume of a Sphere

V = \frac{4}{3} \cdot π \cdot r^{3}

The volume of a sphere is a measure of how much can fit inside the sphere. So for example, a beach ball’s volume tells you how much air needs to be blown into the ball to fill it up. This formula may look a little scary, but it’s easy to use. You need to learn the formula and know how to use it. The volume unit is raised to the power of 3.

Example 3

Find the volume of a sphere with a radius of $2 cm$

V = \frac{4}{3} \cdot π \cdot 2^{3} \approx \frac{4}{3} \cdot 3.14 \cdot 8 \approx 33.49 {cm}^{3}

Rule

Volume $\leftrightarrow$ Liter

1 {dm}^{3} = 1 L

Example 4

How many liters does a sphere with a radius of $1 dm$ hold?

V = \frac{4}{3} \cdot π \cdot 1^{3} \approx 4.19 {dm}^{3}

That is, the sphere holds about $4.19$ L.

Example 5

Find the volume of a hemisphere with a radius of $3 dm$

The first thing we need is the volume of the sphere:

\begin{array}{l} V & = \frac{4}{3} \cdot π \cdot 3^{3} \approx \frac{4}{3} \cdot 3.14 \cdot 27 \\ \approx 113.04 {dm}^{3} \end{array}

V = \frac{4}{3} \cdot π \cdot 3^{3} \approx \frac{4}{3} \cdot 3.14 \cdot 27 \approx 113.04 {dm}^{3}

Then we divide by 2, because we’re only looking for the volume of half the sphere:

\frac{113.04 {dm}^{3}}{2} = 56.52 {dm}^{3}

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What Is the Difference Between a Sphere and an Ellipsoid?

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Surface Area of a Sphere

Surface Area of a Sphere

Volume of a Sphere

Volume of a Sphere

Volume ↔ Liter

Volume $\leftrightarrow$ Liter