An arc of a circle subtends an angle 36° at a point on its circumference. What is the area (in cm2 ) of the sector if the arc of its length is 3∗𝜋5 𝑐𝑚?
Question
An arc of a circle subtends an angle 36° at a point on its circumference.
What is the area (in cm²) of the sector if the arc of its length is (3 \cdot \frac{\pi}{5}\ cm\ ?
Solution
To solve this problem, we need to use the formula for the area of a sector of a circle, which is given by:
Area of sector = (θ/360) * π * r²
Where: θ is the angle the sector subtends at the centre of the circle (in degrees), r is the radius of the circle.
However, we don't know the radius of the circle. But we do know the length of the arc, which is given by:
Arc length = θ/360 * 2πr
We can rearrange this formula to find r:
r = Arc length / (θ/360 * 2π)
Substituting the given values:
r = 3π/5 / (36/360 * 2π) r = 3π/5 / (π/5) r = 3 cm
Now we can substitute r = 3 cm into the formula for the area of a sector:
Area of sector = (36/360) * π * 3² Area of sector = (1/10) * π * 9 Area of sector = 9/10 π cm²
So, the area of the sector is 9/10 π cm².
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