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.

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 ✓.