Circles
3.0 Measurements · 3.1 Circles
Syllabus tag: Kenya CBC | Grade 8 Mathematics | Strand 3.0 Measurements | Sub-Strand 3.1 Circles (5 lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Work out the circumference of a circle
- Work out the length of an arc of a circle
- Calculate the perimeter of a sector of a circle
- Promote the use of circles in real-life situations
Circles
Wrap a string once round a tin and straighten it out. Its length is the circumference of the circle.
Every circle has the same relationship between that distance and the width across, large or small. This topic is about measuring lengths on a circle.
a) The parts you need
The radius is the distance from the centre to the edge.
The diameter goes right across through the centre, so it is twice the radius.
The circumference is the whole distance round the outside.
An arc is a part of the circumference, like a slice of the crust.
A sector is the piece enclosed by an arc and two radii, shaped like a slice of cake.
b) Pi
Divide any circumference by its own diameter and you always get the same number, a little over 3. We call it pi, written π.
For most work use 22/7 or 3.14. Choose 22/7 when the radius is a multiple of 7. The sevens then cancel and the arithmetic stays whole.
c) Circumference
The rule is C = 2πr, or equally C = πd, since the diameter is twice the radius.
Check the radius is really the radius. If the question gives the diameter, halve it first. Using the diameter as the radius doubles every answer, and it is the commonest mistake in this topic.
d) Length of an arc
An arc is a fraction of the circumference. The fraction is the angle at the centre divided by 360.
A 90° angle is a quarter of 360°, so the arc is a quarter of the circumference. 44 divided by 4 is 11.
The same reasoning works for any angle. A 60° arc is one sixth of the way round, because 60 out of 360 is one sixth.
e) Perimeter of a sector
The perimeter of a sector is the whole distance round its edge. That means the curved arc plus the two straight radii.
Forgetting the two radii is the usual slip. A sector is not just the arc. Walk round its edge and you travel along both straight sides too.
f) Length, not area
Everything here measures length, so every answer is in centimetres or metres.
Area of a circle is different, uses a different rule, and gives an answer in square units. That comes in Grade 9. Keep the two apart.
g) Where this is used
A tailor measures round a waist. A wheelwright works out how far a wheel travels in one turn. A road engineer designs a roundabout. A cook edging a round cake with ribbon needs the circumference exactly.
Words to know
- Circumference -- the total distance around a circle, calculated as 2πr or πd.
- Arc -- a portion of a circle's circumference.
- Sector -- the region of a circle bounded by an arc and the two radii at its ends.
:::checkpoint Check yourself
- Find the circumference of a circle of radius 14 cm, taking π as 22/7.
- What fraction of a circle is an arc with a centre angle of 60°?
- A sector has arc 10 cm and radius 6 cm. Find its perimeter.
- Why must you halve the diameter before using C = 2πr? :::
Bridge to practice
Try the exercises below -- on finding circumference, arc length, and the perimeter of a sector.