Area of Part of a Circle
Measurement · Area of Part of a Circle
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 12 Area of Part of a Circle
Lesson objectives
By the end of this topic, you should be able to:
- Find the area of a sector of a circle.
- Find the length of an arc.
- Find the area of a segment.
- Find the area of a common region between two circles.
Area of Part of a Circle
A circle can be divided into parts. Each part has its own area formula, built from the whole.
a) The parts
A sector is bounded by an arc and two radii, like a slice of cake.
A segment is bounded by an arc and a chord, like a slice cut straight across.
A chord is any straight line joining two points on the circle. The longest chord is the diameter.
b) Length of an arc
An arc is a fraction of the circumference.
Arc length = (θ/360) × 2πr, where θ is the angle at the centre.
c) Area of a sector
A sector is that same fraction of the circle's area.
Sector area = (θ/360) × πr².
Both formulas use the same fraction. Only what it multiplies changes: circumference for arc, area for sector.
d) Perimeter of a sector
The perimeter is the arc plus the two radii.
Forgetting the two radii is the standard error. Walk round the edge and you travel along both straight sides.
e) Area of a segment
A segment is what remains when the triangle is cut away from the sector.
Segment = sector area − triangle area.
The triangle has two sides equal to the radius, with the centre angle between them. So its area is ½ r² sin θ.
Work out the sector first, then the triangle, then subtract. Doing it in one step invites mistakes.
f) Common region between two circles
Where two circles overlap, the shared region is made of two segments, one from each circle.
Find the segment from the first circle, then the segment from the second, and add.
Suppose the circles are equal and each passes through the other's centre. The two segments are then identical, so double one.
Sketch the overlap and mark which segment belongs to which circle before calculating.
g) Where this is used
Cutting fabric or sheet metal in curved shapes. Designing arched windows. Working out the area a rotating sprinkler covers. Overlapping signal ranges.
Words to know
- Arc -- part of the circumference of a circle.
- Chord -- a straight line joining two points on a circle.
- Sector -- the region enclosed by two radii and an arc.
- Segment -- the region enclosed by a chord and an arc.
- Subtended angle -- the angle formed at the centre by two radii.
:::checkpoint Check yourself
- Find the area of a 60° sector of a circle of radius 6 cm, taking π as 3.142.
- Find the arc length of that sector.
- What is the difference between a sector and a segment?
- How do you find the area of a segment? :::
Bridge to practice
The exercises begin with sectors and arc lengths, move through perimeters and segments, and finish with working backwards, common regions and composite figures. In every segment calculation, write the sector and the triangle as separate lines before subtracting.