Measuring Space: Perimeter and Area | Exercise 6.3

Question 1

Unless stated otherwise, use the approximation 227\dfrac{22}{7} for π\pi.

Find the area of a sector of a circle with radius 7 cm7\text{ cm} if the angle of the sector is 60∘60^\circ.

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Solution
Understand the Question
  • A sector is a portion of a circular disk enclosed by two radii and an arc connecting them.
  • The area of a sector with central angle θ\theta and radius rr is a fraction of the total circle's area (πr2\pi r^2): Area of sector=θ360∘×πr2\text{Area of sector} = \dfrac{\theta}{360^\circ} \times \pi r^2
  • Substitute r=7 cmr = 7\text{ cm}, θ=60∘\theta = 60^\circ, and π=227\pi = \dfrac{22}{7} into the formula to find the area.

Step 1 · Calculate the Area of the Sector

Given:

  • Radius, r=7 cmr = 7\text{ cm}
  • Angle of sector, θ=60∘\theta = 60^\circ
  • π=227\pi = \dfrac{22}{7}Diagram 1

Area of a sector: Area=θ360∘×πr2\text{Area} = \dfrac{\theta}{360^\circ} \times \pi r^2

Substitute the given values:

Area=60∘360∘×227×72=16×227×49=16×22×7=1546=773\begin{aligned} \text{Area} &= \dfrac{60^\circ}{360^\circ} \times \dfrac{22}{7} \times 7^2 \\[0.6em] &= \dfrac{1}{6} \times \dfrac{22}{7} \times 49 \\[0.6em] &= \dfrac{1}{6} \times 22 \times 7 \\[0.6em] &= \dfrac{154}{6} \\[0.6em] &= \dfrac{77}{3} \end{aligned}
Answer

773 cm2\dfrac{77}{3}\text{ cm}^2

Common Mistakes
  • Confusing Sector Area with Arc Length: Using the formula for arc length (θ360∘×2πr)\left(\dfrac{\theta}{360^\circ} \times 2\pi r\right) instead of the area formula (θ360∘×πr2)\left(\dfrac{\theta}{360^\circ} \times \pi r^2\right).
  • Unit Error: Leaving out units or writing cm\text{cm} instead of cm2\text{cm}^2 for area.
  • Incomplete Simplification: Forgetting to simplify 1546\dfrac{154}{6} to lowest terms 773\dfrac{77}{3}.

More questions in Exercise 6.3

Q1

Unless stated otherwise, use the approximation 227\dfrac{22}{7} for π\pi.

Find the area of a sector of a circle with radius 7 cm7\text{ cm} if the angle of the sector is 60∘60^\circ.

Q2

Find the area of a quadrant of a circle whose circumference is 44 cm44\text{ cm}.

Q3

The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes.

Q4

A chord of a circle of radius 10 cm10\text{ cm} subtends 90∘90^\circ at the centre. Find the area of the corresponding:

(i) minor sector (that subtends 90∘90^\circ at the centre), and (ii) major sector (that subtends 270∘270^\circ at the centre). (Use π≈3.14\pi \approx 3.14.)

Q5

A chord of a circle of radius 15 cm15\text{ cm} subtends an angle of 60∘60^\circ at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use π≈3.14\pi \approx 3.14 and 3≈1.73\sqrt{3} \approx 1.73.)

Q6

A car has two wipers which do not overlap. Each wiper has a blade of length 28 cm28\text{ cm} and sweeps through an angle of 120∘120^\circ. Find the total area cleaned at each sweep of the blades.

Q7

A chord of a circle of radius rr subtends an angle of 60∘60^\circ at the centre of the circle. Show that the area of the corresponding minor segment of the circle is equal to πr2(16−34)\pi r^2 \left( \dfrac{1}{6} - \dfrac{\sqrt{3}}{4} \right).

Q8

An equilateral triangle is inscribed in a circle of radius rr. Show that the ratio of the area of the triangle to the area of the circle is equal to 334π≈0.413\dfrac{3\sqrt{3}}{4\pi} \approx 0.413.

Q9

A square is inscribed in a circle of radius rr. Show that the ratio of the area of the square to the area of the circle is equal to 2π≈0.637\dfrac{2}{\pi} \approx 0.637.

Q10

A hexagon is inscribed in a circle of radius rr. Show that the ratio of the area of the hexagon to the area of the circle is equal to 332π≈0.827\dfrac{3\sqrt{3}}{2\pi} \approx 0.827. Can you see why the answer is exactly twice the answer to Question 8?

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