Measuring Space: Perimeter and Area | Exercise 6.3

Question 2

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

Solution

Let's find the circle's radius first.

Step 1 — Find the radius

We use the circumference formula. The circumference is given as 44 cm.

$2\pi r = 44$

$2 \times \frac{22}{7} \times r = 44$

$\frac{44r}{7} = 44$

$r = \frac{44 \times 7}{44}$

$\boxed{r = 7 \text{ cm}}$

Diagram 1

Step 2 — Calculate the quadrant area

A quadrant is one-fourth of a circle. We use the area formula for a quadrant.

$\text{Area of quadrant} = \frac{1}{4} \times \pi r^2$

$= \frac{1}{4} \times \frac{22}{7} \times (7)^2$

$= \frac{1}{4} \times \frac{22}{7} \times 49$

$= \frac{1}{4} \times 22 \times 7$

$= \frac{154}{4}$

$\boxed{\text{Area of quadrant} = \frac{77}{2} \text{ cm}^2}$

Answer

The area of the quadrant is $\frac{77}{2} \text{ cm}^2$ .

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?