We are given the radius of the circle, r = 10 cm, and the angle subtended by the chord at the center, θ = 60^°. We need to find the area of the minor segment.

First, we calculate the area of the sector formed by the chord. The formula for the area of a sector is (θ)/(360^°) × π r². Substituting the given values, we get (60^°)/(360^°) × 3.14 × (10 cm)².

First, we calculate the area of the sector formed by the chord. The formula for the area of a sector is (θ)/(360^°) × π r². Substituting the given values, we get (60^°)/(360^°) × 3.14 × (10 cm)².

We know that sin(60^°) = (√3)/(2). Using the given value √3 = 1.73, we calculate the area of the triangle as (1)/(2) × 100 × (1.73)/(2) = 50 × 0.865 = 43.25 cm².

(a) Find the area of the minor and the major sectors of a circle with radius 6 cm, if the angle subtended by the minor arc at the centre is $60^\circ$ . (Use $\pi = 3.14$ ) OR (b) If a chord of a circle of radius 10 cm subtends an angle of $60^\circ$ at the centre of the circle, find the area of the corresponding minor segment of the circle. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$ )

Answer: The area of the corresponding minor segment of the circle is $9.08\text{ cm}^2$ .

Step-by-step solution

Step 1: Identify Given Values

We are given the radius of the circle, $r = 10\text{ cm}$ , and the angle subtended by the chord at the center, $\theta = 60^\circ$ . We need to find the area of the minor segment.

Step 2: Calculate Area of Sector

First, we calculate the area of the sector formed by the chord. The formula for the area of a sector is $\frac{\theta}{360^\circ} \times \pi r^2$ . Substituting the given values, we get $\frac{60^\circ}{360^\circ} \times 3.14 \times (10\text{ cm})^2$ .

Step 3: Substitute Values and Compute Sector Area

Simplifying the fraction and performing the multiplication, the area of the sector is $\frac{1}{6} \times 314 = 52.33\text{ cm}^2$ .

Step 4: Calculate Area of Triangle

Next, we calculate the area of the triangle formed by the two radii and the chord. Since the angle subtended at the center is $60^\circ$ and the two sides are radii, the triangle is equilateral. The formula for the area of a triangle with two sides and the included angle is $\frac{1}{2} r^2 \sin\theta$ . Substituting the values, we get $\frac{1}{2} \times (10\text{ cm})^2 \times \sin(60^\circ)$ .

Step 5: Substitute Values and Compute Triangle Area

We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ . Using the given value $\sqrt{3} = 1.73$ , we calculate the area of the triangle as $\frac{1}{2} \times 100 \times \frac{1.73}{2} = 50 \times 0.865 = 43.25\text{ cm}^2$ .

Step 6: Calculate Area of Minor Segment

Finally, the area of the minor segment is the difference between the area of the sector and the area of the triangle. Subtracting the triangle's area from the sector's area gives us the segment's area.

Step 7: Compute Final Segment Area

Subtracting the area of the triangle from the area of the sector, we get $52.33\text{ cm}^2 - 43.25\text{ cm}^2 = 9.08\text{ cm}^2$ .

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