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PSLE Mathematics · Primary 5 and 6

Area and Circumference of Circles

In Primary 6 you find the circumference and area of circles, semicircles and quarter circles, and then use them in composite figures made with squares, rectangles and triangles. Every calculation uses π, and PSLE questions tell you which value to use, usually 3.14 or 22/7, and sometimes the π key on a calculator. Circle questions often carry several steps, so careful setting out matters. This guide explains the formulas, the difference between perimeter and area, a worked composite figure and three practice questions.

What the syllabus expects

  • Identify the centre, radius and diameter of a circle, and use diameter = 2 × radius
  • Find the circumference of a circle using circumference = π × diameter
  • Find the area of a circle using area = π × radius × radius
  • Find the perimeter and area of semicircles and quarter circles
  • Find the perimeter and area of composite figures made of circles, parts of circles and straight-sided shapes
  • Use the value of π given in the question, such as 3.14 or 22/7

Parts of a circle and the value of π

Every point on a circle is the same distance from its centre. This distance is the radius. The diameter is a straight line through the centre from one side to the other, so diameter = 2 × radius. The circumference is the distance all the way round the circle. For every circle, the circumference divided by the diameter gives the same number, called π (pi). π is a little more than 3, and PSLE questions tell you which value to use: 3.14, 22/7, or the π key on a calculator. Use exactly the value stated. If the radius or diameter is a multiple of 7, the question will often ask for 22/7, because the 7 cancels and the arithmetic is neat.

Circumference

Circumference = π × diameter, which is the same as 2 × π × radius. For a circle of diameter 14 cm, taking π = 22/7, the circumference is 22/7 × 14 = 44 cm. For a circle of radius 10 cm, taking π = 3.14, the diameter is 20 cm and the circumference is 3.14 × 20 = 62.8 cm. Circumference is a length, so its unit is cm or m, not cm². A common application is a rolling wheel: in one complete turn, a wheel travels a distance equal to its circumference. A wheel of diameter 70 cm, with π = 22/7, travels 22/7 × 70 = 220 cm in one turn, so it travels 220 × 200 = 44 000 cm = 440 m in 200 turns.

Area

Area of a circle = π × radius × radius. The formula uses the radius, so if you are given the diameter, halve it first. For a circle of radius 7 cm, taking π = 22/7, the area is 22/7 × 7 × 7 = 154 cm². For a circle of radius 10 cm, taking π = 3.14, the area is 3.14 × 10 × 10 = 314 cm². Area is measured in square units such as cm² or m². To find the radius from the circumference, divide by π to get the diameter, then halve it. For example, a circumference of 88 cm with π = 22/7 gives a diameter of 88 ÷ 22/7 = 28 cm and a radius of 14 cm.

Semicircles, quarter circles and composite figures

A semicircle is half a circle, so its area is half the area of the full circle. Its perimeter is the curved part, which is half the circumference, plus the straight diameter. For a semicircle of diameter 20 cm with π = 3.14, the curved part is 1/2 × 3.14 × 20 = 31.4 cm, the perimeter is 31.4 + 20 = 51.4 cm, and the area is 1/2 × 3.14 × 10 × 10 = 157 cm². A quarter circle has a quarter of the area, and its perimeter is a quarter of the circumference plus two radii. For composite figures, find areas by adding or subtracting the area of each piece. For perimeters, trace the outside edge with your finger and add only the lengths you pass. Lines inside the figure are not part of the perimeter.

Worked example

A figure is made up of a square of side 14 cm and a semicircle. The diameter of the semicircle is one side of the square. Taking π = 22/7, find (a) the perimeter of the figure (b) the area of the figure.

  1. The diameter of the semicircle is 14 cm, so the radius is 7 cm.
  2. Curved edge = 1/2 × π × diameter = 1/2 × 22/7 × 14 = 22 cm.
  3. The side shared by the square and the semicircle is inside the figure, so the perimeter uses only the other 3 sides of the square: 3 × 14 = 42 cm.
  4. Perimeter = 42 + 22 = 64 cm.
  5. Area of square = 14 × 14 = 196 cm².
  6. Area of semicircle = 1/2 × π × radius × radius = 1/2 × 22/7 × 7 × 7 = 77 cm².
  7. Area of figure = 196 + 77 = 273 cm².

Answer: (a) 64 cm (b) 273 cm²

Common mistakes

  • Using the diameter in the area formula. Area = π × radius × radius, so halve the diameter first. Using 14 cm instead of 7 cm in the worked example would give an area four times too big.
  • Counting a line inside a composite figure as part of its perimeter. In the worked example, the side shared by the square and the semicircle is not on the outside edge.
  • Forgetting the straight edges of a semicircle or a quarter circle. The perimeter of a quarter circle is a quarter of the circumference plus two radii.
  • Using 3.14 when the question says 22/7, or the other way round. The answers differ slightly, so use exactly the value given.
  • Writing the area in cm or the circumference in cm². Area is in square units and lengths are not.
  • Rounding π × radius halfway through and then multiplying again. Keep exact values until the final step.

Exam tips

  • Write the formula with the stated value of π before substituting, for example 'area = 22/7 × 7 × 7', so the marker can follow your method.
  • Circle the word radius or diameter in the question and write down both values before you start.
  • For perimeter questions, trace the outline of the figure and list each part (curved edges and straight edges) before adding.

Practice questions

1. Easy: Find the circumference of a circle of radius 21 cm. Take π = 22/7.

Show answer

132 cm. Working: diameter = 42 cm, and 22/7 × 42 = 22 × 6 = 132 cm.

2. Medium: A quarter circle has a radius of 10 cm. Taking π = 3.14, find its area and its perimeter.

Show answer

Area 78.5 cm², perimeter 35.7 cm. Working: area = 1/4 × 3.14 × 10 × 10 = 1/4 × 314 = 78.5 cm². Curved edge = 1/4 × 3.14 × 20 = 15.7 cm, and perimeter = 15.7 + 10 + 10 = 35.7 cm.

3. Harder: A rectangular card measures 28 cm by 14 cm. Two circles, each of diameter 14 cm, are cut out of it. Taking π = 22/7, find the area of the card that is left.

Show answer

84 cm². Working: area of rectangle = 28 × 14 = 392 cm². Each circle has radius 7 cm and area 22/7 × 7 × 7 = 154 cm², so the two circles have area 308 cm². Area left = 392 − 308 = 84 cm².

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