Work the area or perimeter backwards and the missing side falls out
From the area back to a side
A rectangle has an area of 24 cm², and one of its sides is 4 cm. How long is the other side? The area was found by multiplying the two sides, so work backwards by dividing.
Think of the rectangle as 4 rows of centimeter squares. 24 squares fill the 4 rows equally, so each row holds 24 ÷ 4 = 6 squares. The other side is 6 cm. Check by multiplying: 6 × 4 = 24.
24 centimeter squares in 4 equal rows: each row holds 6, so the missing side is 6 cm.
From the perimeter back to a side
A rectangle has a perimeter of 26 cm, and one of its sides is 5 cm. A rectangle has two sides of 5 cm, one on each side, so together they use 5 + 5 = 10 cm of the perimeter.
The other two sides share what is left: 26 − 10 = 16 cm. They are the same length, so each is 16 ÷ 2 = 8 cm. Check by walking around: 8 + 5 + 8 + 5 = 26.
The rectangle has two sides of 5 cm and two missing sides of the same length.
The four sides laid end to end along a line: 8, 5, 8 and 5 reach 26. Take away the two 5s, and the two 8s make the 16 that is left.
A square from its area
A square has four equal sides, so its area is the side multiplied by itself. A square has an area of 49 cm². Look for the number that multiplies by itself to make 49. 6 × 6 = 36 is too small, and 8 × 8 = 64 is too big, but 7 × 7 = 49. Each side is 7 cm.
A square from its perimeter is quicker, because the perimeter is shared equally among the four sides. A square with a perimeter of 28 cm has sides of 28 ÷ 4 = 7 cm.
√49 =
Grow the square of tiles from its corner until it holds exactly 49. A side of 6 holds 36, too few, and a side of 8 holds 64, too many. A side of 7 holds 49.
Undo the right step
Each way back undoes the way forward. The area was found by multiplying, so divide it by the side you know; subtracting, as in 24 − 4 = 20, does not give a side. The perimeter holds two of each side, so take off both 5s, not just one. And the area of a square is the side times itself, not the side times 2, so halving 49 does not give its side.
Overlapping shapes · Perimeter of Two Shapes Pushed Together
QuestionTwo identical squares of side 6 cm are pushed together side by side so that they overlap in a strip 2 cm wide and 6 cm tall. Find the perimeter and the area of the combined figure.
1.The combined figure is a rectangle 6 cm tall and 6 + 6 − 2 = 10 cm long.
Two squares of 6 sharing a strip 2 wide, full height. 2.Perimeter = 2 × (10 + 6) = 32 cm.
Together: a rectangle 10 by 6. Perimeter 2 × 16 = 32 cm. 3.Area = 36 + 36 − 2 × 6 = 60 cm2; or 10 × 6 = 60 cm2.
Area: 36 + 36 − 12 = 60 cm². 4.The two squares' own perimeters total 48 cm; the 16 cm lost is the edges now inside the figure.
The two squares had 48 cm of edge; 16 cm of it is now inside.
Answer: 32 cm; 60 cm2
Watch out for
- Adding the two perimeters, 48 cm, as if the squares only touched at a point.
- Subtracting the overlap once from the perimeter, which mixes lengths and areas.
Symmetry and grids · The Perimeter and Area of a Staircase Shape
QuestionA staircase shape fits inside a rectangle 12 cm long and 8 cm high. Its three steps rise from right to left. Going up from the bottom right corner, the edges of the steps are 3 cm up and 4 cm across, then 3 cm up and 4 cm across, then 2 cm up and 4 cm across. (a) Find the perimeter of the shape. (b) Find its area.
1.Walk round the shape. The bottom edge is 12 cm and the left edge is 8 cm. The rest of the walk is up and across the steps.
The bottom edge is 12 cm and the left edge is 8 cm. The rest of the walk is up and across the steps. 2.The flat edges of the steps are 4 + 4 + 4 = 12 cm altogether, as long as the top of the rectangle. The upright edges are 3 + 3 + 2 = 8 cm altogether, as tall as its right side.
The flat step edges make 4 + 4 + 4 = 12 cm and the upright ones 3 + 3 + 2 = 8 cm. 3.(a) So the perimeter is the same as the rectangle's: 12 + 8 + 12 + 8 = 40 cm.
(a) The perimeter is the same as the rectangle's: 12 + 8 + 12 + 8 = 40 cm. 4.The rectangle is 12 × 8 = 96 cm2. Two rectangles above the steps are not part of the shape: above the lowest step, one 4 cm wide and 8 − 3 = 5 cm tall, and above the middle step, one 4 cm wide and 8 − 6 = 2 cm tall.
Two rectangles above the steps are not part of the shape: 4 by 5 and 4 by 2. 5.(b) Take them away: 96 − 4 × 5 − 4 × 2 = 96 − 20 − 8 = 68 cm2. Check: cut the shape into three strips instead, 12 × 3 + 8 × 3 + 4 × 2 = 36 + 24 + 8 = 68 cm2.
(b) The area is 96 − 20 − 8 = 68 cm².
Answer: (a) 40 cm; (b) 68 cm2
Watch out for
- Giving the perimeter as 12 + 8 = 20 cm, the bottom edge and the left edge only. The perimeter is the whole walk round the shape, and the edges of the steps are part of it: they add another 12 + 8 = 20 cm.
- Giving the area as 12 × 8 = 96 cm2, because the perimeter is the same as the rectangle's. The step edges have the same total length as the rectangle's edges, but the spaces above the steps are not part of the shape, so the area is less: 68 cm2.
Your turn
1.A rectangle has area 21 cm² and one side 3 cm. The other side?