A segment of a circle whose radius is 7cm subteds an angel of 135 at the center. Calculate the perimeter of the segment.
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the perimeter is two radii plus the outer arc
arc = 2 * π * r * (135/360)
To calculate the perimeter of the segment, we need to find the length of the arc and the length of the chord.
1. Length of the arc:
The length of an arc can be calculated using the formula:
Length of arc = (angle/360) * 2 * π * radius
In this case, the angle is 135 and the radius is 7cm. Plugging these values into the formula, we get:
Length of arc = (135/360) * 2 * π * 7
Length of arc = (3/8) * 2 * 22/7 * 7
Length of arc = (3/4) * 44
Length of arc = 33 cm
2. Length of the chord:
To find the length of the chord, we need to use the formula:
Length of chord = 2 * radius * sin(angle/2)
In this case, the angle is 135 and the radius is 7cm. Plugging these values into the formula, we get:
Length of chord = 2 * 7 * sin(135/2)
Length of chord = 2 * 7 * sin(67.5)
Length of chord ≈ 2 * 7 * 0.926
Length of chord ≈ 12.954 cm
3. Perimeter of the segment:
The perimeter of the segment is the sum of the length of the arc and the length of the chord:
Perimeter = Length of arc + Length of chord
Perimeter = 33 + 12.954
Perimeter ≈ 45.954 cm (rounded to three decimal places)
Therefore, the perimeter of the segment is approximately 45.954 cm.