Question 5

We will find the area of the minor segment by subtracting the triangle area from the sector area.

Step 1 — Find the area of the minor sector

Let's find the area of the sector first. The radius of the circle is 15 cm. The angle at the centre is 60°. The formula for sector area is $\frac{\theta}{360°} \times \pi r^2$.

$= \frac{60}{360} \times 3.14 \times 15^2$

$= \frac{1}{6} \times 3.14 \times 225$

$= 117.75$

$\boxed{117.75 \text{ cm}^2}$

Step 2 — Find the area of the triangle

The triangle formed by the radii and the chord is isosceles. Since the angle at the centre is 60°, the triangle is equilateral. The side length of the equilateral triangle is 15 cm. The formula for an equilateral triangle's area is $\frac{\sqrt{3}}{4} \times \text{side}^2$.

$= \frac{1.73}{4} \times 15^2$

$= \frac{1.73}{4} \times 225$

$= 97.3125$

$\boxed{97.3125 \text{ cm}^2}$

Step 3 — Find the area of the minor segment

The area of the minor segment is the area of the minor sector minus the area of the triangle.

$= 117.75 - 97.3125$

$= 20.4375$

$\boxed{20.4375 \text{ cm}^2}$

Step 4 — Find the area of the circle

Let's find the total area of the circle. The radius of the circle is 15 cm. The formula for the area of a circle is $\pi r^2$.

$= 3.14 \times 15^2$

$= 3.14 \times 225$

$= 706.5$

$\boxed{706.5 \text{ cm}^2}$

Step 5 — Find the area of the major segment

The area of the major segment is the area of the circle minus the area of the minor segment.

$= 706.5 - 20.4375$

$= 686.0625$

$\boxed{686.0625 \text{ cm}^2}$

Answer

(i) The area of the minor segment is approximately 20.44 cm². (ii) The area of the major segment is approximately 686.06 cm².