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SPI Mathematics · Mensuration 2D

Sectors and Segments

A sector is a slice of a circle cut by two radii; its area and arc are the same fraction of the whole circle as its angle is of 360°.

All Mensuration 2D notes

Why this matters

Four questions in the bank, three MODERATE and one EASY: a sector from its angle, a sector from its perimeter, the area swept by a clock's minute hand, and a shaded region made of segments. One idea settles all four: a sector is the fraction θ/360 of the circle.

Concept 1 of 2: Area of a sector, the clock hand and segments

A slice of pizza with a 90∘90^\circ corner is a quarter of the pizza. A sector's area is the circle's area times its angle over 360∘360^\circ.

Definition

  • Area of a sector =θ360∘×πr2= \frac{\theta}{360^\circ} \times \pi r^2.
  • A clock's minute hand turns 360∘360^\circ in 60 minutes, so 6∘6^\circ a minute. The hour hand turns 30∘30^\circ an hour.
  • A segment is the part between a chord and its arc: segment == sector −- triangle.
  • Shaded regions: build them by adding and subtracting whole shapes you know.

Sector and segment

Asector=θ360∘πr2,Asegment=Asector−AtriangleA_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2, \qquad A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}
  • θangle of the sector at the centre, in degrees

Worked example

Find the area of a sector of radius 14 cm whose angle at the centre is 90∘90^\circ.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Mensuration 2D · Sectors and Segments

The length of the minute hand of a clock is 14 cm. The area swept by the minute hand in 5 minutes is

Minute hand or hour hand

The minute hand turns 6∘6^\circ a minute; the hour hand only 0.5∘0.5^\circ a minute. In 20 minutes the minute hand sweeps 120∘120^\circ, a third of the circle.

Giving the sector when the segment is asked

A segment is the sector minus the triangle formed by the two radii and the chord. A quarter circle of radius 21 cm is 346.5 square cm, but its segment is only 346.5−220.5=126346.5 - 220.5 = 126 square cm.
Drill 2 more on area of a sector, the clock hand and segments

Concept 2 of 2: Arc length and the perimeter of a sector

The curved edge of a sector is the same fraction of the circumference as its area is of the circle. Its perimeter adds the two straight radii to that curve.

Definition

  • Arc length l=θ360∘×2πrl = \frac{\theta}{360^\circ} \times 2\pi r.
  • Perimeter of a sector =2r+l= 2r + l: two radii and the arc.
  • Area from the arc: 12lr\frac{1}{2}lr. This needs no angle at all, so use it when the perimeter is given.

Arc and perimeter

l=θ360∘×2πr,P=2r+l,A=12lrl = \frac{\theta}{360^\circ} \times 2\pi r, \qquad P = 2r + l, \qquad A = \frac{1}{2}lr
  • llength of the arc

Worked example

A sector has radius 21 cm and angle 60∘60^\circ at the centre. Find its arc length, perimeter and area.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mensuration 2D · Sectors and Segments

The perimeter of a sector of a circle of radius 15 cm is 54 cm. Then the Area of the sector will be

Forgetting the second radius

A sector's perimeter has two straight edges. Arc == perimeter −2r- 2r, not perimeter −r- r.

Hunting for the angle

If the arc is known, the area is 12lr\frac{1}{2}lr at once. There is no need to find the angle first.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Area of a sector, the clock hand and segments

    Sector and segment

    Asector=θ360∘πr2,Asegment=Asector−AtriangleA_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2, \qquad A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}
  • Arc length and the perimeter of a sector

    Arc and perimeter

    l=θ360∘×2πr,P=2r+l,A=12lrl = \frac{\theta}{360^\circ} \times 2\pi r, \qquad P = 2r + l, \qquad A = \frac{1}{2}lr

Watch out for (4)

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