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7.07 Volume of spheres

Lesson

Now that we have seen how to find the volume of a cone, let's see if we can relate that to the volume of a sphere.

Exploration

Imagine a sphere that has the same radius as a cone with equal radius and height. How many times greater is the volume of the sphere than the volume of the cone?

Test your conjecture with the applet below. Click the button to pour the water from the cone to the sphere. Then, refill the water in the cone and repeat until the sphere is full.

Then, consider the following:

  1. How many cones of water did it take to fill the sphere?
  2. Recall that the volume of a cone can be found using the formula $V=\frac{1}{3}\pi r^2h$V=13πr2h.  Using formula for the volume of a cone and your answer to the question above, derive the equation for the volume of a sphere.

Notice that if a cone and a sphere have equal radius, and if the cone has a height equal to its radius, then the cone will fill the sphere exactly four times. This means that the volume of the sphere is $4$4 times the volume of the cone.

$\text{Volume of sphere}$Volume of sphere $=$= $4\times\text{Volme of cone}$4×Volme of cone

 

  $=$= $4\times\frac{1}{3}\pi r^2h$4×13πr2h

Substitute the formula for the volume of a cone

  $=$= $4\times\frac{1}{3}\pi r^2r$4×13πr2r

In this special case, $h=r$h=r for the cone.

  $=$= $\frac{4}{3}\pi r^3$43πr3

Simplify

 

Volume of a sphere

 The volume, $V$V, of a sphere can be calculated using the formula

$V=\frac{4}{3}\pi r^3$V=43πr3

where $r$r is the radius of the sphere.

 

Practice questions

Question 1

Find the volume of the sphere shown.

Round your answer to two decimal places.

Question 2

The planet Mars has a radius of $3400$3400 km. What is the volume of Mars?

Write your answer in scientific notation to three decimal places.

Question 3

A ball has a volume of $904.779$904.779 cubic units; what is its radius?

  1. (Give your answer correct to the nearest tenth.)

Outcomes

8.GM.2

Solve real-world and other mathematical problems involving volume of cones, spheres, and pyramids and surface area of spheres.

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