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# Volume of Sphere

## Volume given radius and diameter

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The Volume of Spheres

Learning Goal

By the end of this lesson I will be able to determine the volume of a sphere given its radius.

The Volume of a Sphere

Volume is the measure of the amount of three dimensional space that a figure takes up. Finding the volume of a sphere is a bit complicated because it doesn't have any flat surfaces. To find the volume of spheres, it helps to use pyramids. Imagine a pyramid with its base on the surface of the sphere and its point at the center of the sphere. The radius of the sphere would be the height of the pyramid.

The pyramid makes up a portion of the sphere’s volume. If we can fill the whole sphere with pyramids like this, we would know the volume of the sphere. It would be equal to the volumes of all the pyramids put together. How many pyramids would it take to fill a sphere? That depends on how large the area on the outside of the sphere is. We can combine the surface area of a sphere with the volume formula for a pyramid to calculate the volume of all the pyramids contained within the sphere.

The formula for the volume of a sphere can be given by \begin{align*}Volume = \frac{4\pi r^3}{3}\end{align*}, where r is the radius.

This is how the formula appears on the EQAO formula sheet. [Figure1]

For a more in depth explanation of where the formula comes from, please watch the following video.

Example. Determine the volume of each of the following spheres.

a)

The first step is to identify the radius, which is 6m. From there use the formula to determine the volume. The volume will be measured in cubic units.

\begin{align*}Volume = \frac{4\pi r^3}{3}\end{align*}

\begin{align*}Volume = \frac{4(3.14)(6)^3}{3}\end{align*}   Remember to follow BEDMAS and evaluate \begin{align*}(6)^3\end{align*} first.

\begin{align*}Volume = 904.32m^3\end{align*}

b) A sphere with a diameter of 22cm.

The radius is not given in the question, but the diameter of 22cm is. The radius is half of the diameter, so it measures 11cm. Use this radius measurement in the volume formula.

\begin{align*}Volume = \frac{4\pi r^3}{3}\end{align*}

\begin{align*}Volume = \frac{4(3.14)(11)^3}{3}\end{align*}

\begin{align*}Volume = 5572.45 cm^3\end{align*}

For a review of the concept covered, please watch the following video.

Practice

Directions: Find the volume of each sphere. Round to two decimal places when necessary.

1. A sphere with a radius of 3m.                   (Solution: 113.04m3)
2. A sphere with a radius of 2.5cm.               (Solution: 65.42cm3)
3. A sphere with a radius of 5cm.                  (Solution: 523.33cm3)
4. A sphere with a diameter of 8m.                (Solution: 267.95m3)

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