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5.05 Volume of spheres

Lesson

Volume calculations are useful in a wide range of contexts.  Here are some applications where finding the volume of a sphere would be useful:

  • A toy manufacturer making super bouncy rubber balls would need to know the volume of the ball for manufacturing and packaging of the product.
  • An astronomer who wants to calculate how much the sun weighs would need to know its volume. 
  • A pharmaceutical company making round pills needs to know the dosage of medicine in each pill, which is found using its volume.
  • A party equipment hire company needs to know how much gas fits inside each of their balloons, which is found using its volume.

The volume of a sphere with radius $r$r can be calculated using the following formula:

 

Volume of sphere

Volume of a sphere $=$= $\frac{4}{3}\times\pi\times r^3$43×π×r3
  $=$= $\frac{4}{3}\pi r^3$43πr3

 

If we are asked to find the volume of a hemisphere, we would find the volume of a sphere with the same radius, then halve the result.

 

Worked example

Example 1

Calculate the capacity, to the nearest litre, of a hemispherical bird bath with an internal radius of $0.5$0.5 m.

Think: To calculate the capacity we first need to work out the volume.

The volume of a sphere is given by $V=\frac{4}{3}\pi r^3$V=43πr3. Therefore the volume of a hemisphere is a half of this. Note that we are assuming that the bird bath is full of water. The radius of the hemisphere is $0.5$0.5 m, which is $50$50 cm.

Volume of hemisphere $=$= $\frac{1}{2}\times\text{volume of sphere }$12×volume of sphere
  $=$= $\frac{1}{2}\times\frac{4}{3}\times\pi\times50^3$12×43×π×503
  $=$= $261799.3878$261799.3878... cm3
     
Since $1$1 L $=$= $1000$1000 cm3
Capacity of hemisphere $=$= $\frac{261799.2878}{1000}$261799.28781000
  $=$= $261.7993$261.7993... L
  $=$= $262$262 L (nearest litre)

 

Practice questions

Question 1

Find the volume of the sphere shown.

Round your answer to two decimal places.

Question 2

A sphere has a radius $r$r cm long and a volume of $72\pi$72π cm3. Find the value of $r$r.

  1. Round your answer to two decimal places.

QUESTION 3

Copper weighs approximately $9$9 grams per cubic centimetre. What is the weight of $8$8 solid spheres of copper having a diameter of $16$16 cm?

Give your answer in grams.

  1. Round your answer to the nearest gram.

Outcomes

MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures

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