Area and Perimeter

Perimeter: the distance round the outside

The perimeter of a shape is the total distance round its edge, found by adding the lengths of the sides. Because it is a distance, the answer is in a length unit: cm, m or km. A perimeter is never in cm2.

For a rectangle there is a formula worth knowing: $P=2(l+w)$, where $l$ is the length and $w$ is the width. A rectangle 7 m by 3 m has $P=2(7+3)=20$ m. Adding 7 + 3 + 7 + 3 gives the same answer, and either method scores.

For a triangle, add the three sides. For a parallelogram, add the four sides, which is $2(a+b)$ for the two different side lengths. For a trapezium, add all four sides; there is no shortcut worth memorising, and the parallel sides you use for the area are not enough on their own.

• Rectangle 7 m by 3 m: $P=2(7+3)=20$ m
• Triangle with sides 5 cm, 6 cm and 9 cm: $P=5+6+9=20$ cm
• Parallelogram with sides 8 cm and 5 cm: $P=2(8+5)=26$ cm
• Square of side 9 m: $P=4\times 9=36$ m

Area: why the unit is squared

The area of a shape is the amount of flat surface it covers, measured by counting how many unit squares fit inside it. A rectangle 4 cm by 3 cm holds 12 squares of 1 cm2, so its area is 12 cm2. That is why area is always in square units: the number counts squares, and each of those squares is a square centimetre or a square metre.

Keep the two measures apart in your head. Perimeter is a distance round a line; area is a surface inside a boundary. Two shapes can have the same area and different perimeters: a 6 cm by 4 cm rectangle and an 8 cm by 3 cm rectangle both have an area of 24 cm2, but their perimeters are 20 cm and 22 cm.

If a question gives some lengths in cm and others in m, convert everything into the same unit before you start. An area answer of 24 with the units mixed up is meaningless, and the conversion itself is often worth a mark.

The rectangle and the triangle

The area of a rectangle is $A=l\times w$: length times width. Nothing else is needed, and if the shape is a square both sides are equal, so $A=s^2$.

The area of a triangle is $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is the perpendicular height. The half is there because a triangle is half of a rectangle with the same base and height: two identical triangles fit together to make that rectangle. This is the one formula Cambridge prints for you in the list of formulas.

The height in the formula is measured at right angles to the base, from the base up to the opposite vertex. It is usually drawn as a dashed line inside the shape, with a right-angle mark where it meets the base. A slant side of the triangle is longer than the perpendicular height, and using it gives an answer that is too big.

A triangle with base 12 cm and perpendicular height 9 cm has area $\frac{1}{2}\times 12\times 9=54$ cm2. The same triangle may be drawn turned on its side, in which case a different side becomes the base; the base and the perpendicular height are always a matched pair.

The parallelogram and the trapezium

The area of a parallelogram is $A=bh$: base times perpendicular height. It looks like the rectangle formula and it is, because cutting the triangular end off a parallelogram and sliding it to the other side leaves a rectangle with the same base and the same height.

As with the triangle, the height is measured at right angles to the base, not along the sloping side. A parallelogram with base 7.5 cm, perpendicular height 4.8 cm and slant side 6 cm has area $7.5\times 4.8=36$ cm2. The 6 cm is there to tempt you and is not used in the area.

The area of a trapezium is $A=\frac{1}{2}(a+b)h$, where $a$ and $b$ are the two parallel sides and $h$ is the perpendicular distance between them. Read it as the average of the parallel sides multiplied by the distance between them. A trapezium with parallel sides 9 cm and 15 cm and a distance of 6 cm between them has area $\frac{1}{2}\times (9+15)\times 6=\frac{1}{2}\times 24\times 6=72$ cm2.

The two parallel sides are the ones that never meet. In a diagram they are usually the top and the bottom, but a trapezium can be rotated, so check which pair is actually parallel instead of assuming.

Compound shapes made of rectangles

A compound shape is built from simpler shapes joined together. An L-shape is two rectangles, a T-shape is two rectangles, and a picture frame is a rectangle with a smaller rectangle removed. The rule is to split the shape up or take a piece away, never to guess.

Adding: divide the shape into rectangles with a single straight cut, work out each area and add them. A shape made from a 10 cm by 6 cm rectangle and a 4 cm by 3 cm rectangle has area $60+12=72$ cm2.

Subtracting: if the shape is a rectangle with a piece missing, find the area of the whole rectangle and subtract the missing piece. A 12 cm by 8 cm card with a 5 cm by 3 cm rectangle cut out of a corner has $96-15=81$ cm2 of card left.

For the perimeter of a compound shape, walk round the outside and add only the edges on the boundary. The cut you drew to split the shape is not part of the perimeter, and neither is any edge where two rectangles are joined. On the shape above, the perimeter is 38 cm: 10 cm along the bottom, 6 cm up each end, 6 cm along the exposed top, and then 3 cm, 4 cm and 3 cm round the small rectangle that sticks up.

A T-shape is split the same way: cut it into the horizontal bar and the vertical stem. A bar 14 cm by 4 cm and a stem 4 cm by 9 cm give $56+36=92$ cm2, and walking round the outside of that T gives a perimeter of 46 cm: 14 cm along the top, 4 cm down each end of the bar, 5 cm inwards on each side underneath it, and 5 cm, 4 cm and 5 cm round the stem.

Awkward compound shapes usually hide a length you have to work out before you can find any area. On an L-shape the unlabelled edge is the whole length minus the part you can see, and marking it on the diagram before you calculate is the step students skip when the arithmetic goes wrong.

Working backwards from the area or the perimeter

Questions often give the area and one dimension and ask for the other. Substitute what you know into the formula and solve the equation. A rectangle with area 84 cm2 and width 7 cm gives $7l=84$, so $l=12$ cm, and its perimeter is $2(12+7)=38$ cm.

For a triangle the half has to be dealt with before you divide. A triangle with area 84 cm2 and base 14 cm gives $\frac{1}{2}\times 14\times h=84$, so $7h=84$ and $h=12$ cm. Forgetting the half gives 6 cm, which is exactly half the right answer and a very common slip.

Check every backwards answer by putting it back into the formula. If the perpendicular height is 12 cm, then $\frac{1}{2}\times 14\times 12=84$ cm2, which is the area you were given.

• Area 84 cm2, width 7 cm: length 12 cm, perimeter 38 cm
• Triangle area 84 cm2, base 14 cm: perpendicular height 12 cm
• Square with perimeter 36 m: side 9 m, area 81 m2

The formulas you must know, and the one you are given

Cambridge prints a list of formulas at the front of the paper. The area of a triangle is on it. The perimeter of a rectangle, the area of a rectangle, the area of a parallelogram and the area of a trapezium are not, so they have to be in your memory before you open the paper.

Learn them as words first, then as letters: rectangle, length times width; parallelogram, base times perpendicular height; trapezium, half the sum of the parallel sides times the distance between them; triangle, half the base times the perpendicular height. The words tell you which measurement to reach for when a diagram is labelled with more numbers than the formula needs.

The area of a triangle from two sides and the angle between them, $A=\frac{1}{2}ab\sin C$, is Extended work, and Cambridge examines it with the sine and cosine rules in the trigonometry topic rather than here. If a question gives you two sides and the angle between them, that is the formula it wants, and you will meet it again on the trigonometry page.

Before you substitute anything, check that every length in the formula is in the same unit. A diagram that gives the base in cm and the height in m has to be converted first, or the area comes out a hundred times wrong even though the method was right.