Lesson Guide
Explanation
On a rectilinear shape the unlabelled sides are never a mystery: the sides going across must add to the same total as the full width, and the sides going up must add to the full height. Four plus the missing across side makes seven, so it is three; the missing up side plus three makes five, so it is two. Adding all six sides gives twenty-four. There is also a shortcut worth knowing — an L-shape has exactly the same perimeter as the rectangle it fits inside.
Your Turn
A rectilinear shape is 7 wide and 5 tall. What is its perimeter? Check: 2 × (7 + 5) = 24.
Watch Out
The shortcut works for perimeter, not area. The L-shape has a smaller area than its rectangle.
Explanation
Running the area formula backwards is the commonest reverse on this paper. Rectangle P is twelve by five, an area of sixty square centimetres. Rectangle Q has the same area and is four centimetres wide, so its length is sixty divided by four, fifteen. Check by multiplying back. Two rectangles with equal areas can look completely different, which is exactly what makes the question worth asking.
Your Turn
A rectangle has an area of 60 square centimetres and is 4 cm wide. How long is it? Check: 15 cm.
Watch Out
Divide the area by the side you KNOW of the shape you want. Dividing by a side of the other shape is the trap.
Explanation
For a rectangle, half the perimeter is one long side plus one short side, which makes the reverse quick. A perimeter of thirty-four halves to seventeen, and seventeen take twelve leaves five for the short side. The longer route works too — take both twelve-centimetre sides off the thirty-four, leaving ten for the two short sides, five each — and getting the same answer both ways is the check.
Your Turn
A rectangle has perimeter 30 cm and one side 9 cm. Find the other. Check: 15 − 9 = 6 cm.
Watch Out
Halve the perimeter, not the side. Half of 34 is 17, which is one of each side, not one side.
Explanation
Joining an identical rectangle along one side doubles the area, but the perimeter does not double, because two edges end up inside the new shape. A rectangle w by six has perimeter two w plus twelve; side by side they make a two-w by six shape, with perimeter four w plus twelve. So the perimeter grew by two w — and if the question tells you it grew by sixteen centimetres, then w is eight.
Your Turn
Two identical 5 by 3 rectangles are joined along a side of 3. What is the new perimeter? Check: a 10 by 3 shape, so 26.
Watch Out
The perimeter does not double. Two edges of the join disappear inside the shape.
Explanation
A ring between two nested squares is simply the big square minus the one inside it: five by five take four by four is twenty-five take sixteen, nine. Work inwards and the rings go nine, seven, five, three and one — the odd numbers, which add back to twenty-five, and that total is your check. If the question asks for two shaded rings rather than one, work out each and add. Quiz and exercises now.
Your Turn
What is the area of the ring between a 4 by 4 square and a 3 by 3 square? Check: 16 − 9 = 7.
Watch Out
A ring is a subtraction of two areas. Do not try to count its width around the edge.
📌 Key Points — this lesson in a nutshell
- On a rectilinear shape, the unlabelled sides are the difference of the labelled ones opposite them.
- The across sides add to the same as the whole width; the up sides add to the same as the whole height.
- In reverse: side = area ÷ the other side, and for a rectangle side = half the perimeter − the other side.
- Joining two rectangles adds their areas but the perimeter loses the two joined edges.
Step 1: Introduction
Step 1 of 5
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