Measuring Space: Perimeter and Area | Exercise 6.3

Question 4

A chord of a circle of radius 10 cm subtends 90° at the centre. Find the area of the corresponding:

(i) minor sector (that subtends 90° at the centre), and (ii) major sector (that subtends 270° at the centre). (Use $\pi \approx 3.14$ .)

Solution

We will use the formula for the area of a sector.

Step 1 — Calculate Minor Sector Area

The radius of the circle is 10 cm. The minor sector angle is 90°. Let's use the formula for the area of a sector: $\frac{\theta}{360°} \times \pi r^2$ .

$\text{Area of minor sector} = \frac{90}{360} \times 3.14 \times (10)^2$

$= \frac{1}{4} \times 3.14 \times 100$

$= \frac{1}{4} \times 314$

$\boxed{78.5 \text{ cm}^2}$

Diagram 1

Step 2 — Calculate Major Sector Area

The major sector covers the remaining part of the circle. Its angle is 270°. We use the same area formula with the new angle.

$\text{Area of major sector} = \frac{270}{360} \times 3.14 \times (10)^2$

$= \frac{3}{4} \times 3.14 \times 100$

$= \frac{3}{4} \times 314$

$\boxed{235.5 \text{ cm}^2}$

Answer

(i) The area of the corresponding minor sector is 78.5 cm². (ii) The area of the corresponding major sector is 235.5 cm².

More questions in Exercise 6.3

Unless stated otherwise, use the approximation $\frac{22}{7}$ for $\pi$ .

Find the area of a sector of a circle with radius 7 cm if the angle of the sector is 60°.

Find the area of a quadrant of a circle whose circumference is 44 cm.

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

A chord of a circle of radius 10 cm subtends 90° at the centre. Find the area of the corresponding:

(i) minor sector (that subtends 90° at the centre), and (ii) major sector (that subtends 270° at the centre). (Use $\pi \approx 3.14$ .)

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use $\pi \approx 3.14$ and $\sqrt{3} \approx 1.73$ .)

A car has two wipers which do not overlap. Each wiper has a blade of length 28 cm and sweeps through an angle of 120°. Find the total area cleaned at each sweep of the blades.

A chord of a circle of radius $r$ subtends an angle of 60° at the centre of the circle. Show that the area of the corresponding minor segment of the circle is equal to $\pi r^2 \left( \frac{1}{6} - \frac{\sqrt{3}}{4} \right)$ .

An equilateral triangle is inscribed in a circle of radius $r$ . Show that the ratio of the area of the triangle to the area of the circle is equal to $\frac{3\sqrt{3}}{4\pi} \approx 0.413$ .

A square is inscribed in a circle of radius $r$ . Show that the ratio of the area of the square to the area of the circle is equal to $\frac{2}{\pi} \approx 0.637$ .

Q10

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