GCSE Spheres, cones and pyramids Practice Questions

Write down the formula for the volume of a sphere.

The volume of a sphere depends only on its radius $r$, and the radius is cubed.

Substitute the radius into the formula.

The radius is $3$ cm, so put $r = 3$ into the formula.

State the volume of the sphere.

The volume of the sphere is $36\pi$ $\text{cm}^3$.

Recall the volume formula for a sphere.

The volume of a sphere is $\frac{4}{3}\pi r^3$; the radius is cubed, because a volume is three lengths multiplied together.

Rule out the other options.

The expression $4\pi r^2$ is the SURFACE AREA of a sphere, and so is $\frac{4}{3}\pi r^2$ in shape: both have $r$ squared, so they measure an area, not a volume. The fraction in $\frac{1}{3}\pi r^3$ is the one from the cone formula, and $2\pi r^3$ has the wrong number in front.

Select the correct expression.

The volume of a sphere of radius $r$ is $\frac{4}{3}\pi r^3$.

Write down both volumes.

The cylinder has volume $\pi r^2 h$ and the cone has volume $\frac{1}{3}\pi r^2 h$, with the same $r$ and the same $h$.

Compare the two expressions.

The cone volume is exactly one third of the cylinder volume, whatever the radius and height are.

Test the result with numbers.

A cylinder of radius $3$ cm and height $4$ cm holds $36\pi$ $\text{cm}^3$, and the matching cone holds $12\pi$ $\text{cm}^3$.

Rule out the half and the equality.

One half is the fraction for a triangle, not a cone, and the cone clearly holds less than the cylinder it fits inside.

Rule out the remaining options.

The cone cannot be BIGGER than the cylinder that contains it, so $3V_y$ is impossible; $\frac{2}{3}$ is the fraction for a hemisphere in its cylinder, not a cone.

Select the correct statement.

The cone holds one third of the cylinder with the same base and height.

State the test for a volume.

A length has one dimension, an area two and a volume three. Count the lengths multiplied together in each expression; only three of them can give a volume. Numbers such as $\pi$ and $4$ do not count.

Test the first expression.

This multiplies three lengths together, so it measures a volume. In fact it is the volume of a cylinder.

Test the second expression.

Two lengths multiplied together give an area; this is the curved surface area of a cone.

Test the third expression.

Again two lengths, so this is an area; it is the surface area of a sphere.

Test the fourth expression.

One length only, so this is a length; it is the circumference of a circle.

Test the last expression.

Both terms multiply two lengths, so the sum is an area. It is not a volume, even though it looks complicated.

Check the test with a scaling argument.

Doubling every length multiplies a volume by $8$, an area by $4$ and a length by $2$. Only the first expression grows by a factor of $8$.

Check the units.

A volume must come out in cubic centimetres, and only three lengths multiplied together can do that.

Note why adding is different from multiplying.

Terms can only be added if they measure the same kind of thing, so a sum of two areas is still an area.

Select the expression that could be a volume.

Only $\pi r^2 h$ multiplies three lengths together, so only it can be a volume.

Write down the volume of the cylinder.

The cylinder has radius $r$ and height $2r$, because the sphere must fit exactly inside it.

Simplify the volume of the cylinder.

Multiplying out gives $2\pi r^3$.

Write down the volume of the sphere.

The sphere has radius $r$, so its volume is $\frac{4}{3}\pi r^3$.

Divide one volume by the other.

Both volumes contain $\pi r^3$, so the comparison will not depend on the radius at all.

Simplify the fraction.

Everything cancels except the numbers, and $\frac{4}{3} \div 2 = \frac{2}{3}$.

Test the result with a radius of three.

A sphere of radius $3$ cm has volume $36\pi$ $\text{cm}^3$.

Work out the matching cylinder.

The cylinder round it has radius $3$ cm and height $6$ cm, so it holds $54\pi$ $\text{cm}^3$.

Compare the two numbers.

The sphere fills two thirds of the cylinder, exactly as the algebra said.

Test the result with a radius of six.

A sphere of radius $6$ cm holds $288\pi$ $\text{cm}^3$ and its cylinder holds $432\pi$ $\text{cm}^3$, and the ratio is the same.

Rule out the half.

Half the cylinder would be $27\pi$ $\text{cm}^3$, but the sphere holds $36\pi$ $\text{cm}^3$, so it is more than half.

Rule out the third.

One third of the cylinder is the volume of the CONE that fits inside it, not the sphere.

Rule out the three quarters.

Three quarters of the cylinder is too much: it would leave less empty space than there really is.

Rule out the equality.

The sphere sits inside the cylinder with gaps at the sides, so it cannot have the same volume.

State the general result.

This is Archimedes result: a sphere fills two thirds of the smallest cylinder that contains it, whatever its radius.

Select the correct statement.

The sphere fills exactly two thirds of the cylinder that just contains it.