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Checkpoint 8 Maths · Geometry and Measure · 8Gg.04

Area of Parallelograms and Trapezia

See where the formulae A = bh and A = ½(a + b)h come from, then use them to find areas, missing heights and missing sides in Checkpoint-style problems.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum
  • Use what you know about rectangles, squares and triangles to work out (derive) the formulae for the area of a parallelogram and a trapezium. (8Gg.04)
  • Use the formulae to calculate the area of parallelograms and trapezia, and to find a missing length when the area is known. (8Gg.04)

1. Key words

Word / phraseMeaning / use
areaالمساحة — the amount of flat space inside a shape, in square units (cm², m², mm²)
parallelogramمتوازي أضلاع — two pairs of parallel sides
trapezium (plural: trapezia)شبه منحرف — at least one pair of parallel sides
base (b)القاعدة — the side you measure the height from
perpendicular height (h)الارتفاع العمودي — the distance between the parallel sides, measured at 90°
slant height / sloping sideالضلع المائل — a side that leans; it is NOT the height
deriveيستنتج — show where a formula comes from
formula (plural: formulae)قانون — a rule written with letters

You already know: area of a rectangle = length × width (b × h), area of a square = side × side, and area of a triangle = ½ × base × perpendicular height. Every formula on this page is built from these.

2. Area of a parallelogram

Deriving the formula. Cut a right-angled triangle off one end of the parallelogram (along the dashed height line) and slide it to the other end. The pieces make a rectangle. Nothing was added or removed, so the area has not changed.

hbcutmovebhparallelogramrectangle: area = b × h
A parallelogram can be cut and rearranged into a rectangle with the same base b and the same height h.

The rectangle has length b and width h, so its area is b × h. Therefore:

Area of a parallelogram

Area = base × perpendicular height → A = bh. The height must be measured at right angles to the base, not along the sloping side.

5 cm9 cm6 cm
The height is 5 cm (dashed, at 90° to the base). The sloping side, 6 cm, is not used for the area.

Any side can be the base, as long as you use the height that is perpendicular to *that* side. A useful result: parallelograms (and rectangles) with the same base and the same height always have the same area, however much they lean.

3. Area of a trapezium

A trapezium has two parallel sides (call them a and b) and a perpendicular height h between them. There are two good ways to derive the formula.

Method 1 — two copies make a parallelogram. Take two identical trapezia. Turn one upside down (rotate it 180°) and put it next to the other. Together they make a parallelogram with base (a + b) and height h.

abhtrapeziumabbaparallelogram: base (a + b), height h
Two identical trapezia fit together to make a parallelogram with base a + b.

Area of the parallelogram = (a + b) × h. One trapezium is half of it: area = ½(a + b)h.

Method 2 — split into two triangles. Draw a diagonal. The trapezium becomes two triangles with the same height h: one has base a, the other has base b.

abhyellow triangle: ½ × b × hblue triangle: ½ × a × htotal = ½ah + ½bh= ½(a + b)h
A diagonal splits the trapezium into two triangles with the same height.
Area of a trapezium

Area = ½ × (sum of the parallel sides) × perpendicular height → A = ½(a + b)h. In words: add the parallel sides, multiply by the height, then halve. (You can halve first if that is easier: ½ of 4 = 2, then 2 × 16 = 32.)

القوانين بالعربي
  • متوازي الأضلاع = القاعدة × الارتفاع العمودي (A = bh). لو قصّيت مثلث من ناحية وحطيته في الناحية التانية هيبقى مستطيل.
  • شبه المنحرف = نص × (مجموع الضلعين المتوازيين) × الارتفاع: A = ½(a + b)h. نسختين منه بيعملوا متوازي أضلاع قاعدته a + b.
  • الارتفاع لازم يكون عمودي (زاوية 90°) — الضلع المايل مانستخدموش في المساحة.

4. Method, step by step

  1. Name the shape and write the formula: A = bh or A = ½(a + b)h.
  2. Find the right lengths: for a parallelogram, the base and the perpendicular height; for a trapezium, the two parallel sides (look for arrows) and the height between them. Cross out any sloping length you do not need.
  3. Check the units are the same (all cm or all m). Convert first if they are not.
  4. Substitute and calculate. Show this line: it earns the method mark.
  5. Write square units: cm², m², mm².
  6. Reverse questions (area given, length missing): substitute what you know, then use inverse operations. For a trapezium, multiply the area by 2 first, then divide by the height (or by the sum of the parallel sides).

5. Worked examples

Worked example 1 [2 marks] Work out the area of the parallelogram in section 2 (base 9 cm, sloping side 6 cm, perpendicular height 5 cm).

  1. A = b × h. The height is the perpendicular one: 5 cm.
  2. A = 9 × 5 = 45
  3. Answer: 45 cm²

Common mistake: 9 × 6 = 54 cm². The 6 cm side slopes, so it is not the height.

Worked example 2 [2 marks] Calculate the area of this trapezium.

4 cm10 cm6 cm
Parallel sides 6 cm and 10 cm; perpendicular height 4 cm.
  1. A = ½(a + b)h = ½ × (6 + 10) × 4
  2. = ½ × 16 × 4 = 8 × 4 = 32
  3. Answer: 32 cm²

Common mistake: ½ × 6 + 10 × 4 = 43. Add the parallel sides first (brackets!), then multiply.

Worked example 3 [2 marks] A parallelogram has area 56 cm² and base 8 cm. Work out its perpendicular height.

  1. A = bh, so 56 = 8 × h.
  2. h = 56 ÷ 8 = 7
  3. Answer: 7 cm

Worked example 4 [2 marks] A trapezium has parallel sides 7 m and 13 m. Its area is 60 m². Find the perpendicular height.

  1. 60 = ½ × (7 + 13) × h = ½ × 20 × h = 10h
  2. h = 60 ÷ 10 = 6
  3. Answer: 6 m

Worked example 5 [3 marks] A trapezium has area 45 cm² and height 6 cm. One of its parallel sides is 5 cm. Find the length of the other parallel side.

  1. 45 = ½ × (5 + b) × 6 = 3 × (5 + b)
  2. 5 + b = 45 ÷ 3 = 15
  3. b = 15 − 5 = 10
  4. Answer: 10 cm. Check: ½ × (5 + 10) × 6 = ½ × 15 × 6 = 45 ✓

Worked example 6 [4 marks] A farmer's field is a trapezium. A bag of grass seed covers 50 m² and costs 85 Egyptian pounds. How much will it cost to cover the whole field?

64 m40 m25 mNOT TO SCALE
The field: parallel sides 40 m and 64 m, perpendicular height 25 m (a right-angled trapezium).
  1. Area = ½ × (40 + 64) × 25 = ½ × 104 × 25 = 52 × 25 = 1300 m².
  2. Number of bags = 1300 ÷ 50 = 26 bags (exactly, so no rounding needed).
  3. Cost = 26 × 85 = 2210.
  4. Answer: 2210 Egyptian pounds.

In a right-angled trapezium the height is one of the sides (the side at 90° to both parallel sides). If the number of bags is not a whole number, always round up: you cannot buy part of a bag, and rounding down would leave some of the field uncovered.

Common mistakes
  • Using the sloping side instead of the perpendicular height.
  • For a trapezium: forgetting to halve, or multiplying the two parallel sides together instead of adding them.
  • Using two sides that are not parallel as a and b.
  • Giving the area in cm instead of cm² (area always has square units).
  • Mixing units, for example a base in m and a height in cm. Change to one unit first.
  • In reverse questions, dividing by the height but forgetting to double the trapezium area (or the other way round).
How Checkpoint asks this
  • "Calculate the area of the trapezium/parallelogram." The diagram is often marked NOT TO SCALE and gives extra lengths (sloping sides) you should not use.
  • Reverse questions: "The area is 60 cm². Work out the height." Write the formula with the numbers in, then solve.
  • "Show that…" questions: show every step and finish with the exact number given, e.g. '= 40 cm² as required'.
  • Compound shapes made of rectangles, triangles, parallelograms and trapezia: split, find each area, then add (or subtract).
  • Shapes on a square grid: count squares for the base and the perpendicular height.
  • Explain or derive: "Explain why the area of a parallelogram is base × height." Describe cutting and moving the triangle to make a rectangle.
  • Paper 1 has no calculator: halve the even number first to make the calculation easy.

Practice

Remember
  • Parallelogram: A = base × perpendicular height (it rearranges into a rectangle).
  • Trapezium: A = ½(a + b)h, where a and b are the parallel sides (two copies make a parallelogram; or split into two triangles).
  • Always use the perpendicular height, never the sloping side.
  • Reverse questions: put the numbers into the formula, then use inverse operations.
  • Areas are in square units: cm², m², mm².
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