Find the area of a segment of a circle whose diameter is 16 cm & the central angle measures 80 degrees.

Please include solution. Thanks.

The full circle has an area of pi*R^2. For a segment that has an 80 degree central angle, take 80/360 = 2/9 of the full circle area.

To find the area of a segment of a circle, we need to know the radius and the central angle. Since the diameter of the circle is given as 16 cm, the radius is half of the diameter, which is 16/2 = 8 cm.

The formula for the area of a segment of a circle is (θ/360) * π * r^2, where θ is the central angle and r is the radius of the circle.

In this case, the central angle is given as 80 degrees, so we can substitute the values into the formula:

Area = (80/360) * π * 8^2
= (2/9) * π * 64
= (128/9) * π
≈ 44.96 cm^2

Therefore, the area of the segment of the circle is approximately 44.96 cm^2.

To find the area of a segment of a circle, you need to subtract the area of the corresponding triangle from the area of the sector.

Here's the step-by-step solution:

Step 1: Calculate the radius of the circle. Since the diameter is given as 16 cm, the radius is half of that, which is 16/2 = 8 cm.

Step 2: Calculate the area of the sector. The formula for the area of a sector is: (θ/360) * π * r^2, where θ is the central angle in degrees and r is the radius of the circle. Plugging in the values, we have: (80/360) * π * 8^2 = (2/9) * π * 8^2 = (2/9) * π * 64.

Step 3: Calculate the area of the triangle. The formula for the area of a triangle is: (1/2) * base * height. In this case, the base is one of the sides of the sector, which is the circumference of the circle (2πr), multiplied by the central angle (θ/360), and the height is the radius of the circle (r). Plugging in the values, we have: (1/2) * (2π * 8) * (80/360) * 8 = (π/9) * 8^2 * 2 * 2 = (4π/9) * 64.

Step 4: Subtract the area of the triangle from the area of the sector. (2/9) * π * 64 - (4π/9) * 64 = 128π/9 - 256π/9 = -128π/9.

The negative sign indicates that the area is invalid, which means that the given central angle of 80 degrees is larger than a semicircle (180 degrees) and the segment of the circle doesn't exist.

Therefore, the area of the segment of the circle cannot be calculated for the given central angle of 80 degrees.

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