Lesson Guide
Explanation
The area of a rectangle is the number of unit squares that fit inside it, which is length times width. A six by four rectangle has four rows of six squares, twenty-four square units — twenty-four square centimetres if the units are centimetres. Backwards, an area of sixty with a width of five means a length of twelve. Keep area and perimeter apart: area is the squares inside and comes in square units, perimeter is the distance around and comes in plain units.
Your Turn
Area of a 9 cm by 4 cm rectangle, and of a square of side 8 cm. Check: 36 cm2; 64 cm2.
Watch Out
Area is × , perimeter is +. A 6 by 4 rectangle has area 24 and perimeter 20 — do not swap them.
Explanation
A shape made of rectangles is measured by cutting it into rectangles. This L-shape splits along the dashed line into an upright four by ten, area forty, and a foot six by four, area twenty-four: sixty-four altogether. Cut it the other way — a four by six on top of a ten by four — and you get twenty-four plus forty, the same. Or take the whole ten by ten and subtract the missing six by six corner. Any split with square corners works, the sides of each piece come from the labels, and two different splits agreeing is the best check there is.
Your Turn
An L-shape is made of an 8 by 3 rectangle and a 3 by 4 rectangle. Area? Check: 24 + 12 = 36.
Watch Out
When you split, a side of one piece is often NOT a labelled side — subtract to find it. The foot here is 6 wide because 10 − 4 = 6.
Explanation
When a shape is a rectangle with a rectangle missing from the inside — a frame, a garden with a patch, a path round a pool — the fastest method is whole minus hole. A frame with an outer eight by six and an inner six by four has area forty-eight minus twenty-four, twenty-four. A twelve by six garden with a four by three vegetable patch has seventy-two minus twelve square metres of grass. You could cut a border into four thin rectangles, but one subtraction does it.
Your Turn
Outer rectangle 10 by 8, inner 6 by 4. Area of the frame? Check: 80 − 24 = 56.
Watch Out
Subtract the HOLE's area, not its sides. The frame is 48 − 24, not 8 × 6 − 6 − 4.
Explanation
Nested squares of side six, five, four, three, two, one, stacked in a corner and coloured alternately: the black parts are rings, and each ring is a square minus the next square in. Thirty-six minus twenty-five is eleven, sixteen minus nine is seven, and the innermost black is the two-square, four. Twenty-two black. The white parts come to fourteen, and black plus white is thirty-six, the outer square — which is the check. Write the square numbers out first, then subtract in pairs.
Your Turn
Squares of side 5, 4, 3, 2, 1, alternating from black on the outside. Black area? Check: 25 − 16 + 9 − 4 + 1 = 15.
Watch Out
A ring is outer square MINUS inner square, not outer side minus inner side. 6 × 6 − 5 × 5 = 11, not 1.
Explanation
On a grid, the squares are there to be counted, but for anything bigger than a few squares it is faster and safer to count the sides and multiply. A garden eight by four with a vegetable patch four by two in the middle: patch eight, garden thirty-two, grass twenty-four. Half-squares count as halves. And read the key — if each grid square is four square metres, multiply the count by four. Quiz and exercises now.
Your Turn
A grid rectangle is 5 squares by 3, and each square is 1 m2. Area? Check: 15 m2.
Watch Out
Count squares, not lines. A rectangle 6 lines wide is 5 squares wide.
📌 Key Points — this lesson in a nutshell
- Area of a rectangle is length × width, in square units: cm2, m2.
- A composite shape: split it into rectangles, find each area, add. Any split that uses square corners works.
- A shape with a hole, or a border: whole minus hole. Frame = outer rectangle − inner rectangle.
- Nested squares: shaded rings are outer square − next square in. 6 × 6 − 5 × 5 + 4 × 4 − 3 × 3 + 2 × 2 = 22.
Step 1: Introduction
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