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.
3² = 9 < 49: a square of side 3 holds too few tiles, so √49 is more than 3
Grow the square until it holds 49 tiles: which side? √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.
Worked example: Perimeter of Two Shapes Pushed Together
Question Two 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
Common mistakes
- 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.
Worked example: The Staircase Perimeter
Question A staircase shape fits inside a rectangle 12 cm long and 8 cm high, with its three steps rising from right to left. Going up from the bottom-right corner, 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. Find the perimeter of the shape and its area.
1.The horizontal step edges add to 4 + 4 + 4 = 12 cm, the length of the rectangle; the vertical ones add to 3 + 3 + 2 = 8 cm, its height.
Three steps inside a 12 by 8 rectangle. 2.So the perimeter equals the rectangle's: 2 × (12 + 8) = 40 cm.
Slide each step edge out to the rectangle: the lengths are unchanged. 3.Area: the rectangle is 96 cm2. The cut-out above the first step is 4 cm by 5 cm = 20 cm2; above the second, 4 cm by 2 cm = 8 cm2.
Perimeter = 2 × (12 + 8) = 40 cm. The cut-outs are 4 by 5 and 4 by 2. 4.Area = 96 − 20 − 8 = 68 cm2.
Area = 96 − 20 − 8 = 68 cm².
Answer: 40 cm; 68 cm2
Common mistakes
- Leaving out the step edges when adding the perimeter: they are edges of the shape.
- Using the perimeter trick for the area too; the area does change when steps are cut out.