Calculate the area of a parallelogram using A = base × perpendicular height
Identify and use the perpendicular height (not the slant side) in calculations
Calculate the area of a trapezium using A = ½ × (a + b) × h
Rearrange the trapezium formula to find a missing measurement
Choose the correct formula when given a mixed set of 2D shapes
State answers with correct area units (cm², m², mm²)
Real-World Applications
Architecture: Roof cross-sections are often trapezium-shaped — builders calculate the area to work out how much roofing felt or insulation is needed for pitched roofs and floors.
Engineering: Steel beams and structural supports often have trapezoidal or parallelogram cross-sections; engineers calculate these areas to determine the strength and weight of components.
Landscaping: Garden beds and plots frequently have two parallel sides of different lengths, so the trapezium formula is used to find the area when ordering turf, topsoil, or paving slabs.
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A parallelogram is a four-sided shape where both pairs of opposite sides are parallel (and equal in length). Rectangles and rhombuses are special types of parallelogram.
Area of parallelogram = base × perpendicular height = b × h
Why does A = base × height work?
Cut the triangle off the left end and move it to the right — you get a rectangle with the same base and height.
Common Exam Trap: Slant Height vs Perpendicular Height
⚠️ Warning: Exams often give you both the perpendicular height AND the slant side. Only use the perpendicular height (at 90° to the base). Ignore the slant side!
📄 Worked Example A — Basic parallelogram
A parallelogram has a base of 8 cm and a perpendicular height of 5 cm. Find its area.
Step 1 — Write the formula
A = b × h
Step 2 — Substitute values
A = 8 × 5
Step 3 — Calculate and state units
A = 40 cm²
📄 Worked Example B — Ignoring the slant side
A parallelogram has a base of 12 m, a perpendicular height of 7 m, and a slant side of 8.5 m. Find its area.
Step 1 — Identify the correct measurements
Base = 12 m | Perpendicular height = 7 m | Slant side 8.5 m → IGNORE
Step 2 — Apply formula
A = b × h = 12 × 7
Step 3 — Answer
A = 84 m²
The slant side (8.5 m) is irrelevant — it is a common exam trap. Always use perpendicular height.
✓ Quick Check 1 — Parallelograms
Question 1 of 10
▭ Trapeziums
A trapezium is a four-sided shape with exactly one pair of parallel sides. The two parallel sides are labelled a (shorter, top) and b (longer, bottom). The perpendicular distance between them is the height h.
Area of trapezium = ½ × (a + b) × h where a and b are the parallel sides, h is the perpendicular height
Trapezium = average the parallel sides × height
Step-by-step: a=5, b=11, h=8
💡 Memory aid: "Add the parallel sides, halve it, multiply by the height." Think of it as the average of the two parallel sides multiplied by the height.
📄 Worked Example A
a = 5 cm, b = 9 cm, h = 4 cm. Find the area.
Step 1 — Write the formula
A = ½ × (a + b) × h
Step 2 — Add the parallel sides
a + b = 5 + 9 = 14
Step 3 — Multiply and halve
A = ½ × 14 × 4 = 7 × 4 = 28 cm²
📄 Worked Example B — Finding the height
Area = 40 cm², a = 6 cm, b = 10 cm. Find the perpendicular height h.
Step 1 — Write formula and substitute known values
40 = ½ × (6 + 10) × h → 40 = ½ × 16 × h → 40 = 8 × h
Step 2 — Rearrange to find h
h = 40 ÷ 8 = 5 cm
To rearrange: divide both sides by ½ × (a + b). Here ½ × 16 = 8, so h = 40 ÷ 8 = 5 cm.
✓ Quick Check 2 — Trapeziums
Question 1 of 10
▭ Mixed Problems and Units
Exam questions often mix different 2D shapes. You need to quickly identify the shape, recall the right formula, and apply it correctly. Always check your units at the end.
Shape Formula Reference
Shape
Formula
Key measurements
Common mistake
Rectangle
A = length × width
Two perpendicular sides
Using diagonal instead of side
Triangle
A = ½ × base × height
Base and perpendicular height
Forgetting the ½
Parallelogram
A = b × h
Base and perpendicular height
Using slant side as height
Trapezium
A = ½ × (a + b) × h
Two parallel sides + height
Using non-parallel sides
Units of area are always square units cm → cm² | m → m² | mm → mm² | km → km²
🔎 Exam tips:
1. In a trapezium, first identify which two sides are parallel (marked with arrows or ticks).
2. Always use the perpendicular height — if you see both slant and perpendicular, use perpendicular.
3. Check your answer makes sense — a large shape should have a large area number.
4. If measurements are in metres, area must be in m² — never mix units.
📄 Worked Example — Compound shape
A garden is made up of a rectangle (8 m × 3 m) and a trapezium on top (parallel sides 8 m and 5 m, height 4 m). Find the total area.
Step 1 — Area of rectangle
Arect = 8 × 3 = 24 m²
Step 2 — Area of trapezium
Atrap = ½ × (8 + 5) × 4 = ½ × 13 × 4 = 26 m²
Step 3 — Total area
Total = 24 + 26 = 50 m²
✓ Quick Check 3 — Mixed Problems and Real Life
Question 1 of 10
Practice: Extra Questions
Work out each answer, then type it in and click Check to see if you're right.
#
Question
Your answer
Result
1
Parallelogram: base = 8 cm, height = 5 cm. Area = ?
2
Parallelogram: base = 12 cm, height = 3 cm. Area = ?
3
Trapezium: parallel sides 6 cm and 10 cm, height 4 cm. Area = ½ × (6+10) × 4 = ?
4
Trapezium: parallel sides 5 cm and 9 cm, height 6 cm. Area = ?
5
Parallelogram: base = 7 cm, height = 4 cm. Area = ?
6
Trapezium: parallel sides 3 cm and 7 cm, height 5 cm. Area = ?
7
Parallelogram: base = 9 cm, height = 6 cm. Area = ?
8
Parallelogram: base = 11 cm, height = 5 cm. Area = ?
9
Trapezium: parallel sides 3 cm and 7 cm, height 4 cm. Area = ?
10
Trapezium: parallel sides 6 cm and 10 cm, height 5 cm. Area = ?
11
Trapezium: parallel sides 4 cm and 12 cm, height 6 cm. Area = ?
⚠ Things to Watch Out For
For a parallelogram, use the perpendicular height — not the slant side — with the formula base × height.
For a trapezium, add the two parallel sides (a + b), not any other pair of sides.
Don't forget the ½ in the trapezium formula: Area = ½ × (a + b) × h.
Don't mix up which measurements are the bases and which is the height in a trapezium.
Always give your area in square units (cm², m², etc.).