Key ideas
- The radius (r) goes from the centre to the edge. The diameter (d) goes all the way across through the centre. d = 2 × r and r = d2.
- The circumference is the distance all the way round: C = πd.
- The area is the space inside: A = πr². Area always uses the radius.
- A semicircle is half a circle and a quarter circle is a quarter. Work out the full circle, then ÷ 2 for a semicircle or ÷ 4 for a quarter circle.
- The perimeter of a part circle is the curved edge plus the straight edges. Don't forget them!
- Use π = 3.14. So C = 3.14 × d and A = 3.14 × r × r.
- Area is in square units (cm², m²). Lengths are in cm or m.
Parts of a circle
A circle has a diameter of 18 cm. What is its radius?(1 mark)
Halve the diameter.r = 182r = 9 cm✓1
In plain EnglishThe radius is half of the diameter, because the diameter is two radii joined end to end through the centre.
Circumference
Calculate the circumference of a circle with diameter 15 cm. Use π = 3.14.(2 marks)
Write the formula.C = πd
In plain EnglishCircumference uses the diameter. Writing the formula down doesn't get a mark on its own, but it helps you get the next step right.
Substitute.C = 3.14 × 15✓1
In plain EnglishPut the diameter in place of d.
Calculate.C = 47.1 cm✓2
In plain EnglishMultiply the diameter by 3.14.
Given the radius? Double it first to get the diameter, then use C = πd.
Area
Calculate the area of a circle with diameter 12 cm. Use π = 3.14.(3 marks)
Find the radius.r = 122 = 6 cm✓1
In plain EnglishArea uses the radius, but we were given the diameter. Halve it.
Substitute.A = πr²A = 3.14 × 6²✓2
In plain EnglishSquare the radius only: 6² = 36. The π is not squared.
Calculate.A = 113.04 cm²✓3
In plain EnglishArea is in square units, cm².
⚠ Radius, not diameter. Using 12 instead of 6 gives an answer four times too big.
Semicircles and quarter circles
Calculate the area of a semicircle with radius 5 cm. Use π = 3.14.(2 marks)
Half of a full circle.A = πr² ÷ 2A = 3.14 × 5² ÷ 2✓1
In plain EnglishA semicircle is half a circle, so find the area of the whole circle and halve it.
Calculate.A = 39.25 cm²✓2
In plain EnglishFor a quarter circle you would divide by 4 instead.
Perimeter of part circles
Calculate the perimeter of a quarter circle with radius 8 cm. Use π = 3.14.(3 marks)
The curved edge.Arc = πd ÷ 4Arc = 3.14 × 16 ÷ 4✓1Arc = 12.56
In plain EnglishThe curved edge is a quarter of the full circumference. The diameter of the full circle is 2 × 8 = 16 cm.
Add the straight edges.P = 12.56 + 8 + 8✓2
In plain EnglishA quarter circle has two straight edges, and both are radii: 8 cm each.
Calculate.P = 28.56 cm✓3
In plain EnglishAdd the curved edge and the two straight edges.
⚠ Perimeter means all the way round. For a semicircle, add the diameter. For a quarter circle, add two radii.
Composite shapes
The shape is made from a rectangle 10 cm by 6 cm and a semicircle. Calculate its area. Use π = 3.14.(4 marks)
Split the shape.Rectangle + semicircle
In plain EnglishThe dashed line splits the shape into a rectangle and a semicircle. The semicircle's diameter is the 6 cm side, so its radius is 3 cm.
Area of the rectangle.Rectangle = 10 × 6 = 60 cm²✓1
In plain EnglishLength × breadth.
Area of the semicircle.r = 62 = 3 cmSemicircle = 3.14 × 3² ÷ 2✓2Semicircle = 14.13
In plain EnglishHalf the area of a full circle with radius 3 cm.
Add them.A = 60 + 14.13✓3A = 74.13 cm²✓4
In plain EnglishAdd the two parts.
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Your turn
Calculator allowed: use π = 3.14 (you can leave out the units). Diagrams are not drawn to scale. Stuck? Tap Need a hint?