EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Circles
Area and volume · Lesson 4 of 7
Last Lesson and Before
Answer each one, then check.
1. What is the radius of a circle with diameter 18 cm?
9 cm
2. Work out 5².
25
3. Round 28.274 to 1 decimal place.
28.3
4. Solve r² = 16 (positive answer).
r = 4
Learning Objectives
1. Name the parts of a circle.
2. Work out the circumference of a circle.
3. Work out the area of a circle.
4. Give answers in terms of π.
5. Work backwards from a circumference or area to the radius.
Parts of a Circle
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The diameter is twice the radius: d = 2r. A chord joins two points on the circle; the diameter is the longest chord. A tangent touches the circle at one point. A sector is bounded by two radii and an arc; a segment by a chord and an arc. |
Know every label - the words appear in exam questions without diagrams. |
The Formulae
π (pi) is the number of times the diameter fits around the circumference: 3.14159...
▸ Circumference. C = πd, or C = 2πr.
▸ Area. A = πr² - always the RADIUS, never the diameter.
▸ Order. For πr², square the radius first, then multiply by π.
▸ In terms of π. Leave π in the answer to give it exactly: radius 5 gives area 25π cm².
Circumference
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A circle has diameter 9 cm. Work out its circumference, to 1 decimal place. |
1. Write the formula
C = πd
2. Substitute
C = π × 9 = 9π
3. Calculate
28.274…
Answer: 28.3 cm
Area
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A circle has diameter 15 cm. Work out its area, to 1 decimal place. |
1. Halve the diameter
r = 7.5
2. Write the formula
A = πr² = π × 7.5²
3. Calculate
π × 56.25 = 176.71…
Answer: 176.7 cm²
A Semicircle
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Work out the perimeter of a semicircle with diameter 10 cm. Give your answer (a) in terms of π and (b) to 1 decimal place. |
1. Half the circumference
½ × π × 10 = 5π
2. Add the straight edge, the diameter
5π+ 10
3. Calculate
25.707…
Answer: (a) 5π+ 10 cm (b) 25.7 cm
Working Backwards
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A circle has area 50 cm². Work out its radius, to 2 decimal places. |
1. Write the formula
πr² = 50
2. Divide by π
r² = 50/π = 15.915…
3. Square root
r = 3.989…
Answer: 3.99 cm
Case Study
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CASE STUDY Archimedes and Pi Around 250 BC, the Greek mathematician Archimedes trapped π between two numbers. He drew a regular polygon just inside a circle and another just outside it, and worked out both perimeters - starting with hexagons and doubling the sides again and again up to 96-sided polygons. The circle's circumference had to lie between the two. He proved that π is between 310/71 and 31/7: between 3.1408 and 3.1429. The fraction 22/7 still used today comes from his upper bound. |
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96 sides The polygons Archimedes finally used |
22/7 His upper bound for π, about 3.1429 |
Key Terms
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Radius The distance from the centre to the edge of a circle. |
Diameter A straight line across a circle through its centre; twice the radius. |
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Circumference The distance around the edge of a circle. |
Chord A straight line joining two points on a circle. |
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Tangent A straight line that touches a circle at exactly one point. |
π Pi: the circumference divided by the diameter of any circle, 3.14159... |
Your Task: Measure Pi
12 minutes
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Collect five circular objects - a coin, a mug, a plate, a roll of tape, a bin. Measure the diameter of each with a ruler and the circumference with string. Work out circumference ÷ diameter for each. What do you notice? 1. Measure carefully in mm. 2. Divide circumference by diameter. 3. Compare with your calculator's π. |
A good answer shows: Every ratio should come out close to 3.1, whatever the size. Measuring errors make some a little high or low; the mean of the class's results is usually very close to π.
Can I...?
☐ Name the parts of a circle.
☐ Work out the circumference from the radius or diameter.
☐ Work out the area from the radius or diameter.
☐ Give an answer in terms of π.
☐ Find the perimeter and area of a semicircle.
☐ Find the radius from the circumference or area.
Summary
✓ C = πd = 2πr.
✓ A = πr².
✓ Semicircle perimeter: half the circumference plus the diameter.
✓ Work backwards: divide by π, then (for area) square root.
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EXAM FOCUS A bicycle wheel has diameter 60 cm. How many complete turns does the wheel make when the bicycle travels 1 km? (4 marks) Check whether you've been given the radius or the diameter - and remember area needs the radius. "Give your answer in terms of π" means leave π in: don't multiply it out. |