Units, heights and composite area

Equal lengths do not mean equal heights

A parallelogram of base b and perpendicular height h has area bh. Cutting a triangular end and moving it to the opposite end makes a rectangle with that base and height. The sloping side is not generally the height.

A trapezium with parallel sides a,b and perpendicular separation h has area (1/2)(a + b)h. Two copies fit together into a parallelogram of base a + b and height h, so one has half its area. For parallel sides 6 cm and 10 cm and height 4 cm, the area is 32 cm².

A triangle uses half its base–height rectangle: A = (1/2)bh. Mark the chosen base and perpendicular height before substituting values.

Area units need two length factors

Since 1 m = 100 cm, a square 1 m on each side contains 100 × 100 = 10 000 squares of 1 cm². Thus 1 m² = 10,000 cm², and 2500 cm² = 0.25 m². Dividing by 100 would convert only one direction.

Volume units need three length factors

A cube with edges 1 m has edges 100 cm long. It contains 100 × 100 × 100 = 1 000 000 cubes of volume 1 cm³. Therefore 1 m³ = 1,000,000 cm³. There are three length directions, so the linear factor is used three times.

For example, 250 000 cm³ is 250 000/1 000 000 = 0.25 m³. Dividing by 10,000 would use the square-unit factor for a volume; dividing by 100 would convert only one direction. Check whether the given unit is cm, cm² or cm³ before choosing a factor.

Check your understanding 1 A square-unit factor cannot convert a volume

A claim says that 0.4 m³ is 4000 cm³, using a factor of 10,000. Correct the conversion and explain why that factor is unsuitable.

Your volume conversion and factor explanation, or work on paper

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0.4 × 1 000 000 = 400 000 cm³. The factor 10,000 converts square metres to square centimetres. A cubic metre has 100 centimetre steps along each of three directions, so its cubic-centimetre count uses 100³, not 100².

Partition or subtract, but count each region once

An outer rectangle is 12 m by 8 m. A rectangular 4 m by 3 m corner is removed. The remaining area is 96-12 = 84 m². You can check by splitting the remainder into two non-overlapping rectangles.

Perimeter follows exposed edges, including the two new notch edges. It is not found by subtracting the removed rectangle’s perimeter from the old one; shared and replaced edges have changed roles.

Check your understanding 2 Convert before combining

A 2 m by 1.5 m rectangle has a 50 cm by 40 cm rectangle cut out. Find the remaining area in square metres.

Your answer or drawing on paper

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The outer area is 3 m². The cut-out is 0.5 × 0.4 = 0.2 m². The remainder is 2.8 m². Alternatively 2000 cm² divided by 10,000 gives the same 0.2 m² cut-out.

Do you have enough information for a perimeter?

A triangular corner of known area is removed from a rectangle. If its sloping side length is not given or otherwise determinable, the remaining area may be known while the new perimeter is not. Area and boundary data answer different questions.