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Plane mensuration · Lesson 2 of 3

Circles, arcs and sectors

Circumference and area, then arcs and sectors as a fraction θ/360 of the circle, the perimeter of a sector, and working back to the angle or radius.

15 minYou should already know: Angles, triangles & polygons
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For a circle of radius rr:

circumference=2πrarea=πr2\begin{aligned} \text{circumference} &= 2\pi r \\ \text{area} &= \pi r^2 \end{aligned}
r
CircleC=2πr, A=πr2C = 2\pi r,\ A = \pi r^2
θrarc
Sectorarc =θ360×2πr= \frac{\theta}{360} \times 2\pi r

WAEC questions usually say “take π=227\pi = \frac{22}{7}”. Radii like 7, 14, 21 and 3.5 are chosen so that the 7 cancels nicely.

A sector is a fraction of the circle

A sector is a slice of the circle between two radii, like a slice of cake. Its curved edge is an arc. If the angle at the centre is θ\theta, the sector is the fraction θ360\frac{\theta}{360} of the whole circle. So

arc length=θ360×2πrsector area=θ360×πr2\begin{aligned} \text{arc length} &= \frac{\theta}{360} \times 2\pi r \\ \text{sector area} &= \frac{\theta}{360} \times \pi r^2 \end{aligned}
Arcs and sectorsChange the angle and radius
OAB90°r = 14
90/360fraction of the circle, θ/36022arc = 90/360 × 2πr154area = 90/360 × πr²50perimeter = arc + 2r = 22 + 28
The sector is 90/360 of the whole circle, so its arc is 90/360 of the circumference and its area is 90/360 of the circle's area (with π = 22/7). Its perimeter goes round the arc and both radii: 22 + 14 + 14.

Change the angle and watch the fraction. Then press Trace the perimeter. The edge of a sector is the arc and the two radii:

perimeter of a sector=arc+2r\text{perimeter of a sector} = \text{arc} + 2r

Working backwards

When the area or the arc is given, put it into the formula and solve for the unknown.

Worked example · WAEC 2012

WAEC 2012 · Paper 2 · Q11 (a, b)

A sector of a circle with radius 20 cm20\text{ cm} has an area of 396 cm2396\text{ cm}^2. Calculate, correct to 1 decimal place, the: [Take π=227]\left[\text{Take }\pi = \frac{22}{7}\right]

sectoral angle;

perimeter of the sector;

  1. (a) The angle

    θ360×227×202=396\frac{\theta}{360} \times \frac{22}{7} \times 20^2 = 396

    Make θ\theta the subject:

    θ=396×360×722×400=113.4∘\begin{aligned} \theta &= \frac{396 \times 360 \times 7}{22 \times 400} \\ &= 113.4^\circ \end{aligned}

    Think first. Write the sector-area formula with the numbers you know.

  2. The arc

    • Arc =113.4360×2×227×20{= \frac{113.4}{360} \times 2 \times \frac{22}{7} \times 20}.
    • ≈39.6{\approx 39.6} cm.

    Think first. You need the arc for the perimeter. Use the angle you found.

  3. (b) The perimeter

    Arc plus the two radii: 39.6+20+20=79.639.6 + 20 + 20 = 79.6 cm.

Your turn

WAEC 2011 · Paper 2 · Q3 (a)

  1. (a)

    A sector of a circle with radius 21 cm21\text{ cm} has an area of 280 cm2280\text{ cm}^2. Calculate, correct to 1 decimal place, the perimeter of the sector. [Take π=227]\left[\text{Take }\pi = \frac{22}{7}\right]

Worked solution (try it first)

(a)

  1. Find the angle first: θ360×227×212=280\frac{\theta}{360} \times \frac{22}{7} \times 21^2 = 280, so θ=280×360×722×441\theta = \frac{280 \times 360 \times 7}{22 \times 441}
    ≈72.73∘\approx 72.73^\circ.
  2. Arc =72.73360×2×227×21= \frac{72.73}{360} \times 2 \times \frac{22}{7} \times 21
    ≈26.67\approx 26.67 cm.
  3. (Or use area =12×= \frac12 \times arc × r\times\ r: arc =2×28021≈26.67= \frac{2 \times 280}{21} \approx 26.67 cm.)
  4. Perimeter =26.67+21+21≈68.7= 26.67 + 21 + 21 \approx 68.7 cm.

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