Circles.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Circles

Area and volume

Lesson 4 of 7

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Area and volume

Lesson 4 of 7

Answer each one, then check.

1

What is the radius of a circle with diameter 18 cm?

2

Work out 5².

3

Round 28.274 to 1 decimal place.

4

Solve r² = 16 (positive answer).

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Area and volume

Lesson 4 of 7

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Area and volume

Lesson 4 of 7

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Parts of a Circle

Area and volume

Lesson 4 of 7

Know every label - the words appear in exam questions without diagrams.

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Formulae

Area and volume

Lesson 4 of 7

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Circumference

Area and volume

Lesson 4 of 7

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area

Area and volume

Lesson 4 of 7

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Semicircle

Area and volume

Lesson 4 of 7

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Working Backwards

Area and volume

Lesson 4 of 7

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area and volume

Lesson 4 of 7

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.

96 sides

The polygons Archimedes finally used

22/7

His upper bound for π, about 3.1429

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Area and volume

Lesson 4 of 7

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.

Circumference

The distance around the edge of a circle.

Chord

A straight line joining two points on a circle.

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...

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area and volume

Lesson 4 of 7

YOUR TASK

Measure Pi

12 minutes

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.

WHAT 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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Area and volume

Lesson 4 of 7

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

C = πd = 2πr.

A = πr².

Semicircle perimeter: half the circumference plus the diameter.

Work backwards: divide by π, then (for area) square root.

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.

openrevise.com