GCSE Circles: circumference and area Practice Questions

Write down the formula for the circumference of a circle.

The circumference is $2\pi r$, where $r$ is the radius. The diameter is $d = 2r$, so this is the same as $\pi d$.

Substitute the radius into the formula.

The radius is $5$, so $C = 2\pi \times 5$.

State the circumference.

The circumference is $10\pi\text{ cm}$.

Recall the area formula.

The area of a circle of radius $r$ is $\pi r^2$: the radius is squared and the result is multiplied by $\pi$.

Rule out the expressions that are not areas.

An area needs two lengths multiplied together, so $2\pi r$ and $\pi r$ are the wrong kind of quantity. $(\pi r)^2$ squares $\pi$ as well as $r$, which is about three times too big, and $2\pi r^2$ is exactly twice the area.

Select the correct expression.

The area of a circle of radius $r$ is $\pi r^2$.

Start from the area of the whole circle.

A circle of radius $r$ has area $\pi r^2$.

Halve it.

A semicircle is exactly half of the circle, so its area is half of $\pi r^2$.

Rule out the expression that is a length.

There is no $r^2$ in $\frac{1}{2}\pi r$, so it cannot be an area at all.

Rule out the fractions that are the wrong size.

$\frac{1}{4}\pi r^2$ is a QUARTER circle and $2\pi r^2$ is twice the whole circle, so neither is a semicircle.

Check the answer on a circle of radius 2.

A semicircle of radius $2$ has area $2\pi$, which is half of the $4\pi$ of the whole circle, as it should be.

Select the correct expression.

The area of a semicircle of radius $r$ is $\frac{1}{2}\pi r^2$.

Write down the formula for the circumference of a circle.

The circumference is $2\pi r$, where $r$ is the radius. The diameter is $d = 2r$, so this is the same as $\pi d$.

Work out the circumference.

The circumference is $12\pi$.

Write down the formula for the area of a circle.

The area of a circle of radius $r$ is $\pi r^2$. Only the radius is squared — $\pi$ is not.

Work out the area.

The area is $36\pi$.

Rule out the option with the two answers swapped.

Swapping the circumference and the area is the commonest slip here, and the units give it away: an area cannot be measured in centimetres.

Rule out the option that forgets to double the radius.

$6\pi$ is only half the circumference.

Rule out the option that uses the diameter in the area formula.

Putting the diameter into $A = \pi r^2$ gives $144\pi$, four times too big.

Check the units of each part.

A circumference is a length and an area is a length squared, so only one option has both units right as well as both numbers right.

Check the answers with decimals.

The decimal values confirm that the area is much the bigger number here.

Select the correct statement.

The circumference is $12\pi$ cm and the area is $36\pi$ cm$^2$.

Write down the formula for the circumference of a circle.

The circumference is $2\pi r$, where $r$ is the radius. The diameter is $d = 2r$, so this is the same as $\pi d$.

Use the diameter form of the formula.

The circumference is given as a multiple of $\pi$, so comparing it with $\pi d$ is the quickest route.

Divide both sides by pi.

The $\pi$ cancels straight away, leaving the diameter.

Halve the diameter to get the radius.

The radius is $13$.

Check with the other form of the formula.

Putting $r = 13$ into $C = 2\pi r$ gives back $26\pi$, so the two values are consistent.

Rule out the option with the two answers swapped.

The radius is always the smaller of the two, so a radius bigger than the diameter is impossible.

Rule out the option that doubles the circumference number.

Dividing $26\pi$ by $\pi$ gives $26$, not $52$; that option has multiplied where it should have divided.

Rule out the option with equal radius and diameter.

A diameter is twice the radius, so they can never be equal.

Rule out the option that halves twice.

That option has halved $26$ twice over, once too often.

Work out the area as a further check.

With $r = 13$ the area is $169\pi$, a sensible size for a circle of circumference $26\pi$.

Check the decimal circumference.

The circumference is about $81.68$ cm, a little more than three diameters, as it should be.

Note the mistake to avoid.

Dividing by $2\pi$ gives the radius; dividing by $\pi$ gives the diameter. Mixing the two up is what every wrong option here does.

Check the units.

Both answers are lengths in centimetres, which matches the units of the circumference.

Check the diameter against the circumference.

The circumference divided by the diameter is $\pi$ for every circle, so this pair of values is consistent.

Select the correct statement.

The diameter is $26$ cm and the radius is $13$ cm.