Find the area of the segment AYB shown in Fig. 11.6, if radius of the circle is $21$ cm and $\angle\,\mathrm{AOB} = 120^\circ$. (Use $\pi = \frac{22}{7}$)

Step 1: Calculate the Area of the Sector AOB

To find the area of the segment AYB, we first need to calculate the area of the sector AOB. The formula for the area of a sector with angle $\theta$ and radius $r$ is given by $\frac{\theta}{360^\circ} \times \pi r^2$. Here, the angle $\theta = 120^\circ$ and the radius $r = 21$ cm.

Step 2: Substitute values and compute sector area

Substitute the given values into the formula. We have $\theta = 120^\circ$, $r = 21$ cm, and $\pi = \frac{22}{7}$. Let's perform the calculation: $\text{Area of sector AOB} = \frac{1}{3} \times \frac{22}{7} \times 441 = \frac{1}{3} \times 22 \times 63 = 22 \times 21 = 462 \text{ cm}^2$.

Step 3: Calculate the Area of Triangle AOB

Next, we calculate the area of the triangle AOB. Since OA and OB are radii, $\triangle AOB$ is an isosceles triangle. The formula for the area of a triangle with two sides $r$ and $r$ and included angle $\theta$ is $\frac{1}{2} r^2 \sin\theta$. Here, $r = 21$ cm and $\theta = 120^\circ$.

Step 4: Substitute values and compute triangle area

Substitute the values: $r = 21$ cm and $\theta = 120^\circ$. We know that $\sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$. $\text{Area of } \triangle AOB = \frac{1}{2} \times 441 \times \frac{\sqrt{3}}{2} = \frac{441\sqrt{3}}{4} \text{ cm}^2$.

Step 5: Calculate the Area of the Segment AYB

The area of the minor segment AYB is obtained by subtracting the area of the triangle AOB from the area of the sector AOB. We have already calculated both these areas in the previous steps.

Step 6: Substitute values and compute segment area

Substitute the calculated areas: $\text{Area of segment AYB} = 462 - \frac{441\sqrt{3}}{4} \text{ cm}^2$. This is the exact area of the segment AYB.

More from this chapter

• **Q 9.** A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into
• A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see the figure). Find (i) the area of the part o
• A circle of 6 radius cm is inscribed in a square. Find the area of the shaded region.
• Date: / / A circle of radius 6 cm is inscribed in a square. Find the area of the shaded region.
• A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the minor segment. Use pi = 3.14.
• Find the area of a sector of a circle with radius 4 cm and of angle 30 degrees. Also, find the area of the corresponding major sector. Use pi = 3.14.
• A circle has radius 9 cm. Find its circumference and area. (ref ushj9)
• The radius of a circle is 6 cm. What is its circumference and area?

All Areas Related to Circles solutions →