A sculptor has a block of marble in the shape of a cuboid, with a square base of side 35 cm and a height of 2 m.
He carves the block into a cone, with the same height as the original block and with a base diameter equal to the side length of the original square base.
What is the volume of the marble he removes from the block whilst carving the cone.
Give your answer in m ^{3}, rounded to 3 significant figures.
The volume of the removed material will be equal to the volume of the original marble minus the volume of the cone.
Find the volume of the original marble.
Convert the units of the length, width and height of the cuboid into the same units, either metres or cm.
The question asks for the answer in m ^{3} so it makes sense to use this throughout your calculations.
Length = width = 0.35 m
Height = 2 m
Substitute the values into the formula for the volume of a cuboid.
Find the radius of the base of the cone, this will be half of the diameter.
Find the volume of the cone by substituting the radius and the height into the formula for the volume of a cone.
Find the volume of the marble that was removed by subtracting the volume of the cone from the volume of the cuboid.
Volume removed = 0.245 – 0.0641409 = 0.180859…
Round the answer to 3 significant figures.
Volume of removed marble = 0.181 m ^{3} (3 s.f.)
A doll’s house is in the shape of a prism pictured below. The prism consists of a cuboid with a triangular prism on top of it. The cross section of the triangular prism is an isosceles right-angled triangle. Find the volume of the doll’s house.
Our strategy is to find the area of the triangle and the rectangle and add them together to find the cross-sectional area, and then multiply this by the length to find the volume
As it is an isosceles triangle, length
We can then use Pythagoras to find length
Length will also be
Finding the area of the triangle using
Finding the area of the rectangle
The total cross-sectional area is therefore the triangle plus the rectangle
Finding the volume of the prism by multiplying the cross-sectional area by the length
Rounding to 3 significant figures
79 900 cm ^{3}
The diagram shows a truncated cone (a frustum). Using the given dimensions, find the volume of the frustum.
To find the volume of the frustum, find the volume of the larger cone (30 cm tall, with a radius of 20 cm), and subtract the volume of the smaller cone (15 cm tall, with a radius of 10 cm)
Formula for the volume of a cone:
Calculate the volume of the larger cone
Calculate the volume of the smaller cone
Find the difference
Round to 3 significant figures
11 000 cm ^{3}