Mathematics • Year 8 • Unit 3 • Lesson 6

Area of Parallelograms and Trapezia

Build fluency with A = bh and A = ½(a + b)h. One fully worked example, one guided example with blanks, then eight independent problems ramping from clean parallelograms to find-the-height rearrangements.

Build · I Do / We Do / You Do

1. I do, fully worked example

Read every line. Each step has a short reason so you can see why, not just what.

Problem. A parallelogram has base b = 9 cm and perpendicular height h = 6 cm. Find its area.

Use the perpendicular height, not the slanted side: Area = b × h.

Step 1, Identify the perpendicular height.

h = 6 cm (shortest distance between the two parallel sides, at 90°).

Reason: the formula only works with the perpendicular height, never the slant side.

Step 2, Write the formula.

A = b × h

Reason: this is the parallelogram formula. It comes from rearranging the shape into a rectangle.

Step 3, Substitute and calculate.

A = 9 × 6 = 54

Reason: base times perpendicular height, same as a rectangle with the same b and h.

Step 4, State with units.

A = 54 cm²

Reason: area is always in square units (cm² here, because the sides are in cm).

Answer: A = 54 cm².

Stuck? Revisit lesson § Card 5, slice the triangle off one end of the parallelogram, attach to the other → rectangle with base b and height h.

2. We do, fill in the missing steps

Same shape as Section 1, but this time for a trapezium. Fill in each blank. 4 marks

Problem. A trapezium has parallel sides a = 5 cm and b = 11 cm, and perpendicular height h = 7 cm. Find the area.

Step 1, Write the formula:

A = ______ × (a + b) × h

Step 2, Add the parallel sides:

a + b = 5 + 11 = ______ cm

Step 3, Substitute and calculate:

A = ½ × ______ × 7 = ______ × 7 = ______

Step 4, State with units:

A = ______ cm²

Stuck? Revisit lesson § Card 6, two trapezia put together form a parallelogram of base (a + b), so one trapezium is half that.

3. You do, independent practice

Show all working. The first three are foundation (direct substitution). The middle three are standard (decimals, mixed shapes). The last two are extension (find an unknown side).

Foundation, direct substitution

3.1 Parallelogram with b = 12 cm, h = 7 cm. Find A. 1 mark

3.2 Parallelogram with b = 15 cm, h = 8 cm. Find A. 1 mark

3.3 Trapezium with a = 4 cm, b = 10 cm, h = 6 cm. Find A. 1 mark

Standard, mixed shapes, decimals

3.4 Trapezium with a = 9 m, b = 15 m, h = 6 m. Find A. 2 marks

3.5 Parallelogram with b = 2.5 m, h = 1.4 m. Find A. 2 marks

3.6 Trapezium with parallel sides 7.5 cm and 12.5 cm and perpendicular height 4 cm. Find A. 2 marks

Extension, find the unknown

3.7 A parallelogram has area 96 cm² and base 12 cm. Find its perpendicular height h. (Hint: rearrange A = bh → h = A ÷ b.) 2 marks

3.8 A trapezium has area 60 cm², parallel sides a = 8 cm and b = 12 cm. Find h. (Hint: 60 = ½(8 + 12)h = 10h, so h = ?) 2 marks

Stuck on 3.7 / 3.8? Use the same formula but solve for h. For 3.7: h = 96 ÷ 12. For 3.8: simplify the right-hand side first.

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Answers, Do not peek before attempting

Section 2, We do (trapezium 5, 11, 7)

Step 1: A = ½ × (a + b) × h.
Step 2: a + b = 5 + 11 = 16 cm.
Step 3: A = ½ × 16 × 7 = 8 × 7 = 56.
Step 4: A = 56 cm².

3.1, Parallelogram b = 12, h = 7

A = 12 × 7 = 84 cm².

3.2, Parallelogram b = 15, h = 8

A = 15 × 8 = 120 cm².

3.3, Trapezium a = 4, b = 10, h = 6

A = ½ × (4 + 10) × 6 = ½ × 14 × 6 = 7 × 6 = 42 cm².

3.4, Trapezium a = 9, b = 15, h = 6 (m)

A = ½ × (9 + 15) × 6 = ½ × 24 × 6 = 12 × 6 = 72 m².

3.5, Parallelogram b = 2.5, h = 1.4 (m)

A = 2.5 × 1.4 = 3.5 m².

3.6, Trapezium 7.5, 12.5, h = 4

A = ½ × (7.5 + 12.5) × 4 = ½ × 20 × 4 = 10 × 4 = 40 cm².

3.7, Find h, parallelogram

h = A ÷ b = 96 ÷ 12 = 8 cm. Check: 12 × 8 = 96 ✓.

3.8, Find h, trapezium

60 = ½ × (8 + 12) × h = ½ × 20 × h = 10h, so h = 60 ÷ 10 = 6 cm. Check: ½ × 20 × 6 = 60 ✓.