The radius of circle is 20cm .find .1- lenght of arc, 2- area of sector, 3-area of segment whose central angle is 120degree?

Question

Answer

Answer

Answer

Length of arc is 41.90
area of arc Is 419.04

Answer

1. Length of Arc:
The length of an arc can be calculated using the formula:

Length of Arc = (θ/360) x 2πr

Where θ is the central angle and r is the radius of the circle.

In this case, the central angle is 120 degrees and the radius is 20 cm. Plugging in these values into the formula, we get:

Length of Arc = (120/360) x 2π(20)
= (1/3) x 2π(20)
= (2/3)π(20)
= (40/3)π
≈ 41.89 cm

Therefore, the length of the arc is approximately 41.89 cm.

2. Area of Sector:
The area of a sector can be calculated using the formula:

Area of Sector = (θ/360) x πr²

Again, θ represents the central angle and r is the radius of the circle.

Using the given values of a central angle of 120 degrees and a radius of 20 cm, we can substitute these values into the formula:

Area of Sector = (120/360) x π(20)²
= (1/3) x π(400)
= (1/3) x 400π
≈ 418.88 cm²

Therefore, the area of the sector is approximately 418.88 cm².

3. Area of Segment:
To find the area of a segment, we first need to find the area of the corresponding sector and then subtract the area of the triangle formed by the radii and the chord.

The area of the sector with a central angle of 120 degrees and a radius of 20 cm is already calculated as approximately 418.88 cm².

Next, we need to find the area of the triangle. In this case, the triangle is an equilateral triangle because the central angle is 120 degrees. The base of the equilateral triangle is the chord, which is a side of the segment. The height can be found using the formula:

Height = sqrt(3) x (side length of triangle)/2

Since the side length of the triangle is equal to the radius (20 cm), we can substitute this value into the formula:

Height = sqrt(3) x (20)/2
= sqrt(3) x 10
= 10√3 cm

Now, we can find the area of the equilateral triangle by using the formula:

Area of Triangle = (base x height)/2

Substituting the values, we get:

Area of Triangle = (20 x 10√3)/2
= 200√3/2
= 100√3 cm²

Finally, we can find the area of the segment by subtracting the area of the triangle from the area of the sector:

Area of Segment = Area of Sector - Area of Triangle
= 418.88 cm² - 100√3 cm²
≈ 318.88 cm² - 173.21 cm²
≈ 145.67 cm²

Therefore, the area of the segment with a central angle of 120 degrees is

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