Hard GCSE Circles: circumference and area 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 $\pi d$.

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.

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.

Put the given area into the formula.

The area is $60$, so $\pi r^2 = 60$.

Divide both sides by pi.

This gives $r^2 = 19.09859\ldots$

Take the square root.

The radius is $4.37019\ldots$, taking the positive root.

Round to 1 decimal place.

$4.37019\ldots$ rounds to $4.4$.

Check by working forwards again.

Squaring the unrounded radius and multiplying by $\pi$ returns the $60$ that was given.

Rule out the option that is the diameter.

Doubling the radius gives about $8.7$, which is the DIAMETER, not the radius.

Rule out the option that forgets the square root.

Stopping at $r^2$ gives about $19.1$, which is the radius SQUARED.

Rule out the option that divides by 2 pi.

Dividing the area by $2\pi$ treats it as if it were a circumference, which it is not.

Check the size of the radius with a rough estimate.

Taking $\pi$ to be about $3$ predicts a radius near $4.47$. The true $\pi$ is a little LARGER than $3$, so dividing by it leaves a little less: the radius should come out a little below $4.47$, and it does.

Check the units.

The area was in square centimetres and the radius is a length in centimetres.

Note the danger of rounding too early.

Rounding $r^2$ before square rooting it can easily move the first decimal place of the answer.

State the exact form behind the decimal.

The decimal is only a rounded version of this exact value.

Check the rounded radius reproduces the area.

The rounded radius gives an area close to $60$, which confirms it is the right option.

Select the correct radius.

The radius is $4.4$ cm to 1 decimal place.

Split the shape into two pieces.

The square and the semicircle do not overlap, so their areas add.

Write down the area of the square.

A square of side $s$ has area $s^2$.

Find the radius of the semicircle.

The side of the square is the DIAMETER of the semicircle, so the radius is half of it.

Write down the area of the whole circle.

Squaring $\frac{s}{2}$ gives $\frac{s^2}{4}$, so the full circle has area $\frac{1}{4}\pi s^2$.

Halve it to get the semicircle.

A semicircle is half of the circle, so its area is $\frac{1}{8}\pi s^2$.

Add the two areas.

This is the total area of the shape.

Rule out the option that forgets to halve the circle.

The circle on diameter $s$ has area $\frac{1}{4}\pi s^2$; the semicircle is half of that.

Rule out the option that uses the side as the radius.

Taking $r = s$ instead of $r = \frac{s}{2}$ makes the circle four times too big.

Rule out the option that halves the wrong circle.

This halves a circle of radius $s$, not a circle of radius $\frac{s}{2}$, so it is four times too big as well.

Rule out the option that is not an area.

There is no $s^2$ in $2\pi s$, so it cannot be added to $s^2$ at all.

Test the expression on a square of side 4.

A square of side $4$ carries a semicircle of radius $2$, whose area is $\frac{1}{2}\pi \times 2^2 = 2\pi$. The expression agrees.

Test the expression on a square of side 2.

A square of side $2$ carries a semicircle of radius $1$, of area $\frac{1}{2}\pi$. The expression agrees again.

Check the expression is dimensionally consistent.

Both terms contain $s^2$, so the two pieces can be added; an expression mixing $s$ and $s^2$ could not be an area.

Write the answer with a common factor.

Taking $s^2$ out shows that the semicircle adds about $39\%$ to the area of the square, whatever the value of $s$.

Select the correct expression.

The total area is $s^2 + \frac{1}{8}\pi s^2$.

Split the perimeter into the curved edge and the straight edge.

The outline of a semicircle is half a circumference plus the diameter across the bottom.

Write the curved edge in terms of the radius.

Half of $2\pi r$ is $\pi r$.

Write the straight edge in terms of the radius.

The diameter is twice the radius.

Form an equation for the perimeter.

The two parts together make the perimeter, $36$.

Factorise the left hand side.

Both terms contain $r$, so it can be taken out as a factor. This is the step that makes the equation solvable in one move.

Divide by the bracket.

$\pi + 2 = 5.1415\ldots$, so $r = 7.00172\ldots$

Check by working the perimeter out again.

Putting the radius back into $\pi r + 2r$ gives $36$, as required.

Check the size of the answer with a rough estimate.

Since $\pi + 2$ is a little MORE than $5$, dividing by $5$ instead gives a little too much: the radius should come out a little below $7.20$, and it does.

Work out the diameter as well.

Doubling the radius gives the straight edge of the semicircle.

Work out the curved edge as well.

The curved edge is a little more than three radii long.

Check the two parts add to the perimeter.

The curved edge and the diameter add back to the perimeter given.

Note the mistake to avoid.

Dividing the perimeter by $\pi$ treats the shape as if it had no straight edge, and gives a radius that is far too big.

Note the danger of rounding too early.

The value of $\pi + 2$ is irrational; rounding it before dividing can move the last digit of the answer.

Check the units.

The perimeter was a length in centimetres, so the radius is too.

Round the answer to 1 decimal place.

Rounding $7.00172\ldots$ to 1 decimal place gives $7.0$.

Trace the outside of the quarter circle.

The outline is made of two radii and one quarter of the circumference.

Add the two straight edges.

Both straight edges are radii, so together they measure $22$.

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 $\pi d$.

Take a quarter of the circumference.

The whole circumference is $22\pi$, so the curved edge is $5.5\pi$.

Add the curved edge to the straight edges.

The perimeter is the two radii plus the arc.

Replace pi by its decimal value.

Using $\pi = 3.14159\ldots$ gives $39.27875\ldots$, which is where the calculator answer comes from.

Check the size of the answer by taking pi to be roughly 3.

Replacing $\pi$ by $3$ gives about $38.5$. The perimeter should be a little larger than this, because $\pi$ is a little larger than $3$.

Note the mistake to avoid.

The arc on its own is not the perimeter: a quarter circle is a closed shape with two straight edges as well.

Check the units of the answer.

A circumference is a length, so it is measured in centimetres, not in square units.

Check the arc against the straight line joining its ends.

The straight distance between the ends of the arc is $15.55\ldots$, and the curved route must be longer.

Check by fitting four of them together.

The four arcs make the whole circumference back again.

Write the answer as a single exact expression.

This is the same answer written before the arithmetic is finished off.

Write the perimeter in factorised form.

Taking out the common factor $11$ shows that the perimeter is always about $3.57$ radii, whatever the size of the quarter circle.

Work out the area of the quarter circle as well.

The same radius gives the area; a perimeter and an area are different kinds of quantity and must not be mixed up.

Round the answer to 2 decimal places.

Rounding $39.27875\ldots$ to 2 decimal places gives $39.28$.