Perimeter of Shapes with Mixed Units

1. Perimeter and the units we measure with

The perimeter of a plane shape is the distance all the way round its edge. If you walked once round the edge of a football pitch, the distance you walked is the perimeter of the pitch. For any shape with straight sides we find it by adding the lengths of all the sides.

For a rectangle with length $l$ and width $w$ there are two sides of each length, so the perimeter is $P=l+w+l+w=2(l+w)$. For a square of side $s$ it is $P=4s$. For a triangle, we simply add its three sides.

The one rule you must never break is this: you can only add measurements that are in the same unit. Adding 3 metres to 40 centimetres by writing 43 is meaningless, just as adding 3 cedis to 40 pesewas to get 43 is wrong. Before adding we must make all the units match.

▶ Moving to a smaller unit means multiplying (more, smaller pieces). Moving to a bigger unit means dividing. Watch the arrows turn round in the last step.

2. Method 1: convert everything to the smaller unit

The first method is to change every side into the smaller unit so that all the numbers are whole numbers. For a shape with sides in km and m, we change everything to metres. For sides in m and cm we use centimetres, and for sides in cm and mm we use millimetres.

1. 1Write down every side of the shape.
2. 2Convert each side to the smaller of the two units, e.g. 2 km 300 m = 2,000 m + 300 m = 2,300 m.
3. 3Add all the converted sides (or use $2(l+w)$ for a rectangle).
4. 4If the answer is large, change it back to the bigger unit, e.g. 6,200 m = 6.2 km.

Take a triangular plot with sides 2 km 300 m, 1 km 850 m and 2 km 50 m. In metres the sides are 2,300 m, 1,850 m and 2,050 m. Their sum is 6,200 m. Since 1,000 m make 1 km, the perimeter is 6.2 km.

3. Method 2: convert to a decimal in the larger unit

The second method writes each side as one decimal number in the larger unit. Because 1 cm = 10 mm, each millimetre is one tenth of a centimetre, so 7 cm 5 mm = 7.5 cm. Because 1 m = 100 cm, each centimetre is one hundredth of a metre, so 1 m 75 cm = 1.75 m.

Once every side is a decimal in the same unit we add them, lining up the decimal points. Look at Maame Esi's plot in the animation. Each step shows one move in the working.

▶ Notice that the answer, 42.5 m, is in metres because we converted everything to metres first.

4. Choosing a method and checking your work

Both methods give the same answer, so choose the one that feels safer. For km and m, learners usually prefer converting to metres. For cm and mm or m and cm, the decimal method is quick and the answer is already neat. Whichever you pick, write the unit beside every number so that you never mix them up.

Real problems often ask for more than the perimeter. A farmer may need to know how many rounds of wire go on the fence, or what the fencing will cost. In those cases find the perimeter first, then multiply by the number of rounds or by the price per metre.

• Check the units in the final answer: perimeter is a length, so it is in cm, m or km, never cm² or m².
• Estimate first: 12 m 50 cm is about 12.5 m and 8 m 75 cm about 9 m, so half the perimeter is about 21 m and the perimeter about 42 m. Our answer of 42.5 m is sensible.
• For irregular shapes, tick each side as you add it so no side is left out or counted twice.

Worked examples

Try each one yourself first, then open the solution and compare every step.