Measuring Space: Perimeter and Area | Exercise 6.3

Question 2

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

Solution

Understand the Question

The circumference of a circle is given by $2\pi r$ .
A quadrant is one-fourth ( $\frac{1}{4}$ ) of a circle, with an area given by $\dfrac{1}{4}\pi r^2$ .
We first determine the radius $r$ using the given circumference ( $44\text{ cm}$ ), and then calculate the area of the quadrant.

Step 1 · Find the Radius

Given circumference of the circle $= 44\text{ cm}$ . Diagram 1

\begin{aligned} 2\pi r &= 44 \\[0.6em] 2 \times \dfrac{22}{7} \times r &= 44 \\[0.6em] \dfrac{44r}{7} &= 44 \\[0.6em] r &= \dfrac{44 \times 7}{44} \\[0.6em] r &= 7\text{ cm} \end{aligned}

Step 2 · Calculate the Area of the Quadrant

A quadrant is one-fourth of a circle.

\begin{aligned} \text{Area of quadrant} &= \dfrac{1}{4} \times \pi r^2 \\[0.6em] &= \dfrac{1}{4} \times \dfrac{22}{7} \times (7)^2 \\[0.6em] &= \dfrac{1}{4} \times \dfrac{22}{7} \times 49 \\[0.6em] &= \dfrac{1}{4} \times 22 \times 7 \\[0.6em] &= \dfrac{154}{4} \\[0.6em] &= \dfrac{77}{2}\text{ cm}^2 \end{aligned}

Answer

$\dfrac{77}{2}\text{ cm}^2$

Common Mistakes

Forgetting the Quadrant Factor: Calculating the area of the entire circle ( $\pi r^2 = 154\text{ cm}^2$ ) instead of multiplying by $\dfrac{1}{4}$ .
Incorrect Formula: Confusing the circumference formula ( $2\pi r$ ) with the area formula ( $\pi r^2$ ) when solving for radius $r$ .
Units Error: Writing the final unit as $\text{cm}$ instead of $\text{cm}^2$ for area.

More questions in Exercise 6.3

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

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

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

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 $10\text{ cm}$ subtends $90^\circ$ at the centre. Find the area of the corresponding:

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

A chord of a circle of radius $15\text{ cm}$ subtends an angle of $60^\circ$ 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 $28\text{ cm}$ and sweeps through an angle of $120^\circ$ . Find the total area cleaned at each sweep of the blades.

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

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 $\dfrac{3\sqrt{3}}{4\pi} \approx 0.413$ .

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 $\dfrac{2}{\pi} \approx 0.637$ .

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 $\dfrac{3\sqrt{3}}{2\pi} \approx 0.827$ . Can you see why the answer is exactly twice the answer to Question 8?