Area & Perimeter of 2-D Shapes — Grade 9 Maths (CAPS)

Area and perimeter formulae for squares, rectangles, triangles, parallelograms, trapeziums and circles, plus composite shapes, unit conversions and what happens to area when the dimensions double.

Also known as: trapezoid area · composite shapes · compound area

Perimeter is the distance around the outside of a shape — measured in mm, cm or m. Area is the space inside it — measured in square units. Mixing up the units is the fastest way to lose a mark.

1The formulae you must know

The area formulae you need. Both the triangle and the parallelogram use the perpendicular height.

Square: A = s² · P = 4s
Rectangle: A = l × b · P = 2(l + b)
Triangle: A = ½ × b × h
Parallelogram: A = b × h
Trapezium: A = ½(a + b) × h
Rhombus or kite: A = ½ × d₁ × d₂ (the diagonals d₁ and d₂ split it into triangles)
Circle: A = πr² · C = 2πr

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Notice that the parallelogram and triangle both use the perpendicular height, not the slanted side. Using the slant is the most common error in this topic.

2Triangles and parallelograms

Worked ExampleA triangle has base 12 cm and perpendicular height 7 cm. Find its area

1
A = ½ × b × h.
Write the formula first.
2
A = ½ × 12 × 7 = ½ × 84.
3
A = 42 cm².
Area is always in square units.

Worked ExampleA parallelogram has base 9 cm and height 5 cm. Find its area

1
A = b × h = 9 × 5.
No halving — a parallelogram is two triangles.
2
A = 45 cm².

3Trapeziums

Worked ExampleA trapezium has parallel sides 8 cm and 12 cm, height 5 cm. Find its area

1
A = ½(a + b) × h.
Average the two parallel sides, then multiply by the height.
2
A = ½(8 + 12) × 5 = ½(20) × 5.
3
A = 10 × 5 = 50 cm².

4Circles

Worked ExampleA circle has radius 7 cm. Find its area and circumference (π ≈ 3,14)

1
A = πr² = 3,14 × 7² = 3,14 × 49.
Square the radius BEFORE multiplying by π.
2
A = 153,86 cm².
3
C = 2πr = 2 × 3,14 × 7 = 43,96 cm.
Circumference uses 2πr, not πr².

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If the question gives the diameter, halve it first. Using d in place of r makes the area four times too big.

5Composite shapes

Break the shape into rectangles, triangles and parts of circles, work out each area, then add (or subtract for a cut-out).

Worked ExampleA 10 cm by 8 cm rectangle has a 3 cm by 2 cm rectangle cut out. Find the remaining area

1
Whole rectangle: 10 × 8 = 80 cm².
2
Cut-out: 3 × 2 = 6 cm².
3
Remaining area = 80 − 6 = 74 cm².
Subtract because the piece was removed.

6Unit conversions and doubling

1 cm² = 100 mm² · 1 m² = 10 000 cm²
Double every length → the perimeter doubles but the area becomes 4 times bigger.

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Area scales by the square of the scale factor. Enlarge by 3 and the area becomes 3² = 9 times as big.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
A square has an area of 225 cm². Determine the length of one side, and hence its perimeter.
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side = √225 = 15 cm (2)
Perimeter = 4 × 15 = 60 cm (1)
2
Area of a rectangle 9 cm by 6 cm.
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54 cm²
3
Perimeter of a rectangle 9 cm by 6 cm.
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2(9 + 6) = 30 cm
4
Area of a triangle, base 12 cm, height 7 cm.
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½ × 12 × 7 = 42 cm²
5
Area of a parallelogram, base 9 cm, height 5 cm.
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45 cm²
6
Area of a trapezium, parallel sides 8 cm and 12 cm, height 5 cm.
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½(20) × 5 = 50 cm²
7
Area of a circle with radius 7 cm (π ≈ 3,14).
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3,14 × 49 = 153,86 cm²
8
Circumference of a circle with radius 7 cm (π ≈ 3,14).
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2 × 3,14 × 7 = 43,96 cm
9
Area of a circle with diameter 10 cm (π ≈ 3,14).
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r = 5 → 3,14 × 25 = 78,5 cm²
10
A 10 × 8 rectangle has a 3 × 2 piece cut out. Find the remaining area.
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80 − 6 = 74 cm²
11
A shape is enlarged by scale factor 3. What happens to its area?
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It becomes 9 times bigger (3²).