The base of a hemisphere has an area of 32𝜋 centimetres squared. Work out the volume of the hemisphere, giving your answer accurate to two decimal places.
Let’s remind ourselves that a hemisphere is simply half of a sphere and would look something like this. We’re told that the base of this hemisphere has an area of 32𝜋 square centimetres and the shape that’s formed on the base of the hemisphere will be a circle. We can recall that the formula for the area of a circle is equal to 𝜋𝑟 squared, where 𝑟 is the radius. Therefore, for the circle of the base of the hemisphere with an unknown radius of 𝑟, we can write the formula the area of the circle is equal to 𝜋𝑟 squared.
And since we’re given that the area is equal to 32𝜋 square centimetres, we can substitute this in to give us 32𝜋 equals 𝜋𝑟 squared. Cancelling the 𝜋 from both sides of our equation will give us that 32 is equal to 𝑟 squared. To find 𝑟 then, we take the square root of both sides, giving us the square root of 32 is equal to 𝑟. And so, we’ve established that the radius of this hemisphere is root 32. We can now use this to help us find the volume of the hemisphere.
We will use the formula to find the volume of a sphere, which tells us that this is equal to four-thirds 𝜋𝑟 cubed. Since this is the formula for a sphere and not a hemisphere, that means when we find our volume, we must halve it. So using the formula, the volume of a sphere equals four-thirds 𝜋𝑟 cubed. And substituting our value 𝑟 equals root 32 gives us that the volume equals four-thirds 𝜋 times root 32 cubed. We then use our calculator to evaluate this, taking care just to take the third part of root 32 and not the whole equation. We can then write that the volume is equal to 758.252 and so on cubic centimetres.
And now, to find the volume of the hemisphere, we need to halve the value for the volume of the sphere, which gives us 379.126 and so on cubic centimetres. And finally, since we’re asked to give our answer accurate to two decimal places, we check the third decimal digit to see if it is five or more. As it is, then our answer for the volume of the hemisphere will round up to 379.13 cubic centimetres.