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Teach Areas Related to Circles in five steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

  • Equal pizza slices share the circular surface equally.
  • Edging a sector-shaped tabletop requires measuring its arc and both straight sides.
  • A circular garden path covers the outer circle’s area minus the inner circle’s area.

What to teach

Parts of a circle

Sector, segment and quadrantCircle pieces get their names from their boundaries.Area of a sectorThe angle determines the share of the circle’s area.

Curved edges and remaining areas

Length of an arc and perimeter of a sectorAn arc measures the curve; perimeter includes the straight edges.Area of a segmentSubtract the triangle from its sector to leave the minor segment.Areas of combinations of plane figuresAdd separate pieces, or subtract excluded parts, to find an area.

How to teach it

Parts of a circle

Three circles show a sector, a segment and a quadrant with their boundaries labelled.

1. Sector, segment and quadrant

Circle pieces get their names from their boundaries.

Show
Look at the diagram’s shaded pieces from left to right. Trace their straight boundaries, then find the right-angle mark.
Ask
If a chord replaces the two radii, what will the enclosed piece be called?
Say
A sector has two radii and a curved arc. A segment has a chord and an arc. A quadrant is a sector with a right angle.
Used in
Circular window designs use sectors and segments to specify the glass pieces needed.
If your child says…“Every piece with a curved edge is a sector.”
Ask
Does the middle piece have two radii or one chord?
What’s true
The straight boundaries distinguish a sector from a segment.
Equal circles compare a shaded quadrant with a sector of another central angle.

2. Area of a sector

The angle determines the share of the circle’s area.

Show
In the diagram, find the quarter, then radius r and angle θ\theta in degrees.
Ask
The right angle gives a quarter; what share does the other marked angle give?
Say
Equal angles cover equal areas within a circle. Multiply the whole area by the angle’s share of 360°: θ360πr2\frac{\theta}{360}\pi r^2.
Used in
A clock hand’s length and turn determine the sector it sweeps, helping measure the swept area.
If your child says…“A quarter-circle needs a quarter of the radius.”
Ask
Does the quadrant reach the same outer edge as the whole disc?
What’s true
The quadrant keeps the full radius but covers only a quarter of the area.

Curved edges and remaining areas

A sector’s boundary follows one radius, its curved arc and the other radius.

3. Length of an arc and perimeter of a sector

An arc measures the curve; perimeter includes the straight edges.

Show
Find radius r, angle θ\theta in degrees, and arc length ℓ\ell in the diagram.
Ask
How far is it around this sector?
Say
The arc takes the angle’s share of the circumference: ℓ=θ360 2πr\ell=\frac{\theta}{360}\,2\pi r. Add both radii for the perimeter.
Used in
Binding a quadrant-shaped table mat requires its full perimeter, so every edge gets covered.
If your child says…“The curved edge is the whole perimeter.”
Ask
Which edges would remain bare if binding covered only the curve?
What’s true
Both straight edges belong to the boundary too.
A chord divides a quadrant into a blue triangle and a gold minor segment.

4. Area of a segment

Subtract the triangle from its sector to leave the minor segment.

Show
Look at the diagram’s quadrant. Find the chord, then the triangle below it and the curved cap above it.
Ask
Would using the whole quadrant’s area overestimate or underestimate the gold cap’s area?
Say
It would overestimate the cap’s area. Subtract the triangle’s area from the quadrant’s area. Here, the perpendicular radii give the triangle’s base and height.
Used in
A window’s segment-shaped glass needs its area measured to estimate the glass required.
If your child says…“The curved cap has the whole quadrant’s area.”
Ask
Which coloured region would you count extra by using the whole quadrant?
What’s true
The quadrant contains both the cap and the triangle.
Concentric circles show a shaded ring and both radii measured from their shared centre.

5. Areas of combinations of plane figures

Add separate pieces, or subtract excluded parts, to find an area.

Show
In the diagram, find the shared centre. Follow outer radius R and inner radius r, then point to the shaded ring.
Ask
If only the inner circle grows, will the shaded ring’s area increase or decrease?
Say
The shaded ring’s area will decrease. Subtract the inner disc’s area from the outer disc’s area: πR2−πr2\pi R^2-\pi r^2.
Used in
A circular path’s ring area determines how much paving material is needed.
If your child says…“The ring’s width is the radius for its area.”
Ask
Does the ring’s width reach from the shared centre to either curved edge?
What’s true
Ring width measures the gap between circles, not either circle’s radius.

A real problem

Outer sprinklers cover more land

Zybach’s centre-pivot irrigation system was patented in 1952.

The machine rotates around a fixed central point.

With equally spaced sprinklers, those farther out cover larger rings and usually use larger nozzles.

Workers could leave the system assembled during planting, cultivation and harvesting.

Smithsonian MagazineNC State UniversityNorth Dakota State University

Centre-pivot sprinkler arm watering a green crop field.

Concept Lab · Common misconception

The tempting mistake in Areas Related to Circles

Open standalone misconception page →

Does half the radius give half the area?

Wrong

A circle of radius 3 has half the area of one of radius 6.

Why it feels right. The radius is halved.

Right

Area depends on the square of the radius.

Worked example

The larger circle has twice the radius. Its area is four times as large.
  1. The areas are 9π and 36π square units.
  2. The smaller circle covers one quarter of the larger area.

Ask your child five quick questions.

One per step. The answer and the reason are given.

  1. What is the smaller piece between an off-centre chord and the curved edge called?

    A minor segment. Its straight boundary is a chord, rather than two radii.

  2. A disc has radius 7 cm. Four equal sectors are cut from it. What is each sector’s radius?

    7 cm. Quartering the disc changes its area, while its radius stays unchanged.

  3. A quadrant’s arc measures 11 cm and its radius 7 cm. What is its perimeter?

    25 cm. Add both 7 cm straight edges to the 11 cm arc.

  4. A quadrant covers 154 square centimetres. Its triangle covers 98. What area remains beyond the chord?

    56 square centimetres. Subtract the triangle because it lies outside the curved cap.

  5. A ring has radii 15 m and 13 m. Which circle’s area must be removed?

    The inner circle’s area. Removing the inner disc from the outer disc leaves precisely the ring.

15 more questions with answers, 1 with pictures