An arc has a measure of 40 degrees, and its endpoints are at (1,5) and (5,3). What's the area of the circle?

Question

Please post your answer and your reasoning. Thanks!

Answer 1

Let

c = chord length
r = radius of circle
θ = central angle of circle = 40°
A = area of circle

First, solve for c.

c = √{(5-1)² + (3-5)²} = √(4² + 2²) = √20 = 2√5
c/2 = √5

sin(θ/2) = (c/2)/r
sin 20° = (√5)/r
r = (√5)/sin 20° = 6.5378254

A = πr² = π{(√5)/sin 20°}² = 5π/sin² 20° = 134.2816

Answer 2

Interesting problem!

First, calculate the length (l) of the cord between (1,5) and (5,3):

l = √20 = 2√5

half of the length (l) / radius (r) = sin 20 (used symmetry)

Solve for r, r = √5/sin 20

area = pi r^2 = pi (√5/sin 20)^2 = 134.28 units^2

Answer 3

Same answer as northstar.

Answer 4

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