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Circle Geometry: Arc Length and Sector Area

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An automated agricultural sprinkler sweeps across a section of a circular field, forming a circular sector. The area of the irrigated sector is 45π45\pi square meters, and the length of the outer boundary arc of the sector is 6π6\pi meters. What is the radius, in meters, of the circular field?

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Answer: 15

Answer

15 meters
Using the relationship between sector area, arc length, and radius, A=12rsA = \frac{1}{2} r s, we substitute A=45πA = 45\pi and s=6πs = 6\pi to get 45π=12r(6π)=3πr45\pi = \frac{1}{2} r (6\pi) = 3\pi r. Solving for rr gives r=15r = 15 meters.

Step-by-Step Solution

1
Recall the formulas for sector area (AA) and arc length (ss) in terms of radius (rr) and central angle (θ\theta in radians).
s=rθs = r\theta and A=12r2θA = \frac{1}{2}r^2\theta
These are the fundamental geometric formulas for circular sectors.
2
Express the sector area formula in terms of arc length ss.
A=12r(rθ)=12rsA = \frac{1}{2} r (r\theta) = \frac{1}{2} r s
Substituting s=rθs = r\theta simplifies the calculation by eliminating the central angle θ\theta.
3
Substitute the given values A=45πA = 45\pi and s=6πs = 6\pi into the simplified formula and solve for rr.
45π=12r(6π)  ⟹  45π=3πr  ⟹  r=1545\pi = \frac{1}{2} r (6\pi) \implies 45\pi = 3\pi r \implies r = 15
Dividing both sides by 3π3\pi yields the radius r=15r = 15 meters.

Key Concept

Relationship between Sector Area, Arc Length, and Radius
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