Area of Basic Shapes
Five formulas, one strategy: identify the shape, find the perpendicular height, substitute — and always write the unit. This lesson covers rectangles, triangles, parallelograms, trapeziums and circles, then shows you how to break apart composite shapes using addition or subtraction.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
You are tiling a bathroom floor. The floor is L-shaped — it is not a simple rectangle. How would you figure out how many tiles you need? What would your strategy be?
Without calculating — write your gut strategy. We'll revisit this at the end of the lesson.
Every area problem in Maths Standard uses one of five core formulas. The key to composite shapes is knowing which formula fits which part.
All area formulas multiply two lengths together — which is why area is always in square units. The perpendicular height $h$ is always measured at 90° to the base. Never use the slant side as $h$.
Key facts
- The area formula for rectangles, triangles, parallelograms, trapeziums and circles
- What a composite shape is
- The correct units for area answers
Concepts
- Why different shapes need different formulas — and what each part measures
- How to break a complex shape into simpler parts to add or subtract
- Why area is always expressed in square units
Skills
- Calculate the area of any standard shape by selecting and applying the correct formula
- Find the area of composite shapes, including those requiring subtraction
- Write every area answer with the correct unit
Before memorising any formula, it helps to understand where it comes from. Every area formula connects back to the most basic one: the rectangle.
Rectangle: $A = \ell w$ — This is the foundation. A rectangle with length 5 cm and width 3 cm contains exactly 15 unit squares. Area counts those squares.
Triangle: $A = \tfrac{1}{2}bh$ — A triangle is exactly half a rectangle with the same base and height. The $\tfrac{1}{2}$ is always there.
Parallelogram: $A = bh$ — Slice a triangle off one end and reattach to the other — you get a rectangle with the same base and height.
Trapezium: $A = \tfrac{1}{2}(a + b)h$ — The formula averages the two parallel sides ($\tfrac{1}{2}(a + b)$), then multiplies by the height. Think of it as a rectangle with an average width.
Circle: $A = \pi r^2$ — The area depends on the square of the radius. Double the radius and the area quadruples — not doubles.
The perpendicular height problem
The single most common error across all of these formulas is using the slant side instead of the perpendicular height. The height $h$ must be measured at 90° to the base.
What to write in your book
- All five formulas: $A = \ell w$ · $A = \tfrac{1}{2}bh$ · $A = bh$ · $A = \tfrac{1}{2}(a+b)h$ · $A = \pi r^2$
- $h$ = perpendicular height (dotted line, meets base at 90°). Never use the slant side.
- All area formulas multiply two lengths together — hence answers are always in square units.
True or false: In the formula $A = \tfrac{1}{2}bh$, the $h$ can be the slant side of the triangle if that is the labelled side in the diagram.
A composite shape is any shape that is not one of the five standard shapes. The strategy is always the same:
- Sketch the shape and identify the parts
- Decide: are you joining (add) or removing (subtract)?
- Write your plan: e.g. "Area = rectangle + triangle" before touching numbers
- Calculate each part separately
- Combine the areas and state the unit
Area = area of square − area of circle.
You cannot see both shapes independently — one is missing — but you calculate both and subtract.
Calculator use for circle problems
Using 3.14 instead of the $\pi$ button introduces rounding error early — and that error compounds through the rest of your calculation.
When the question gives you the diameter, write $r = d \div 2$ as a separate line before substituting into $A = \pi r^2$. Substituting the diameter directly gives an answer four times too large (because $(2r)^2 = 4r^2$).
What to write in your book
- Composite shapes strategy: sketch → decide (add or subtract) → write plan → calculate each part → combine.
- Write the plan "Area = rectangle + triangle" before any numbers — earns method marks.
- Circle: always halve the diameter to get $r$ before substituting. Write $r = d \div 2$ as a separate line.
- Use the $\pi$ button — never 3.14. Round only at the very final step.
Quick check: A circle has a diameter of 10 cm. What is the correct area, to 2 decimal places?
Worked examples · 3 in a row, reveal as you go
Find the area of a triangle with base 9 cm and perpendicular height 6 cm.
Triangle: $A = \tfrac{1}{2}bh$
$A = \tfrac{1}{2} \times 9 \times 6$
$A = \tfrac{1}{2} \times 54 = 27$
$A = 27\text{ cm}^2$
Find the area of a circle with diameter 14 cm. Give your answer correct to 2 decimal places.
$r = 14 \div 2 = 7\text{ cm}$
$A = \pi r^2 = \pi \times 7^2 = \pi \times 49$
$A = 153.9380...$
$A = 153.94\text{ cm}^2$
A rectangular piece of timber is 20 cm long and 12 cm wide. A semicircle is cut from one end, with a diameter equal to the width of the rectangle. Find the area of the remaining shape, correct to 2 decimal places.
Area = rectangle $-$ semicircle
diameter $= 12$ cm, so $r = 6$ cm
$A_{\text{rect}} = 20 \times 12 = 240\text{ cm}^2$
$A_{\text{semi}} = \tfrac{1}{2}\pi r^2 = \tfrac{1}{2} \times \pi \times 36 = 18\pi$
$= 240 - 18\pi = 240 - 56.548...$
$= 183.45\text{ cm}^2$
What to write in your book
- Trapezium: $A = \tfrac{1}{2}(a + b)h$ — common error is forgetting the $\tfrac{1}{2}$, giving double the correct answer.
- Composite: write the plan first ("= rectangle − semicircle"), then calculate each part, then combine.
- Keep $\pi$ exact until the very last step — this avoids rounding errors accumulating.
Fill the gap: A trapezium with parallel sides 5 m and 11 m and perpendicular height 4 m has area $A = \tfrac{1}{2}(5 + 11) \times 4 = \tfrac{1}{2} \times 16 \times 4 = $ m².
Common errors · the 3 traps that cost marks
What to write in your book
- Always check: is the height given the perpendicular height or the slant side?
- For every circle: write $r = d \div 2$ as a separate line before substituting.
- For every composite shape: decide add or subtract before calculating. Write it as your first line.
Odd one out: Three of these are correct statements about area formulas. Which one is wrong?
Quick-fire practice · 5 calculations
Find the area of a rectangle with length 13 cm and width 7 cm.
Find the area of a triangle with base 10 m and perpendicular height 6 m.
Find the area of a circle with radius 5 cm. Give your answer to 2 decimal places.
Find the area of a trapezium with parallel sides 4 cm and 9 cm, and perpendicular height 6 cm.
A shape is made from a rectangle (8 cm × 5 cm) with a triangle (base 8 cm, perpendicular height 3 cm) on top. Find the total area.
Top 3 list: List THREE key steps you would take when calculating the area of any composite shape.
Earlier you described how you'd tile an L-shaped floor. Now you know the strategy: split the L into two rectangles, calculate the area of each using $A = \ell w$, and add them together. That is exactly the composite shape method. You had the right idea.
What has changed? What did you get right? What surprised you?
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
SA 1. Find the area of a trapezium with parallel sides 7 cm and 13 cm and a perpendicular height of 8 cm. (2 marks)
1 mark correct substitution, 1 mark correct answer with unit.
SA 2. A logo is made from a square of side 6 cm with a circle of diameter 6 cm inscribed inside it. Find the area of the square that is not covered by the circle, correct to 2 decimal places. (3 marks)
1 mark radius, 1 mark both areas, 1 mark correct subtraction and rounding.
SA 3. A garden is in the shape of a rectangle 12 m × 8 m. A triangular garden bed with base 4 m and perpendicular height 3 m is cut from one corner, and a semicircular garden bed of diameter 8 m is added along one long edge.
(a) Find the area of the triangle removed. (1 mark)
(b) Find the area of the semicircle added, correct to 2 decimal places. (1 mark)
(c) Find the total area of the garden, correct to 2 decimal places. (2 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: $A = 13 \times 7 = $ 91 cm² · 2: $A = \tfrac{1}{2} \times 10 \times 6 = $ 30 m² · 3: $A = \pi \times 25 = $ 78.54 cm² · 4: $A = \tfrac{1}{2}(4+9) \times 6 = $ 39 cm² · 5: Rectangle $= 40$ cm²; Triangle $= 12$ cm²; Total = 52 cm²
SA 1 (2 marks): $A = \tfrac{1}{2}(7 + 13) \times 8 = \tfrac{1}{2} \times 20 \times 8 = $ 80 cm² [1 substitution, 1 answer].
SA 2 (3 marks): $r = 6 \div 2 = 3$ cm [1]. Square: $6^2 = 36$ cm²; Circle: $\pi \times 9 = 28.27$ cm² [1]. Remaining: $36 - 28.27 = $ 7.73 cm² [1].
SA 3 (4 marks): (a) $A = \tfrac{1}{2} \times 4 \times 3 = $ 6 m² [1]. (b) $r = 4$ m; $A = \tfrac{1}{2}\pi \times 16 = 8\pi = $ 25.13 m² [1]. (c) Total $= (12 \times 8) - 6 + 8\pi = 96 - 6 + 25.13 = $ 115.13 m² [2].
Five timed questions on area of basic and composite shapes. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering questions on area of basic shapes. Pool: lessons 1–2.
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