A chord with a length of #8 # runs from #pi/8 # to #pi/6 # radians on a circle. What is the area of the circle?

Question

A chord with a length of #8   # runs from #pi/8   # to #pi/6    # radians on a circle. What is the area of the circle?

Caroline Aucoin · Accepted Answer

#text{Area}= pi( 1 + 64/ ( sqrt(2 + sqrt(2 + sqrt(3))) )) #

We have an isosceles triangle, apex at the center of a circle, sides #r#, #r#, and #8#, central angle #theta = pi/6-pi/8=pi/24.#

Law of Cosines:

# 8^2 = r^2 + r^2 - 2 (r)(r) cos theta = 2(r^2-1) cos theta#

#r^2 = 1 + 32/cos (pi/24) #

Our answer is #pi# times that. Are we done?

No, #pi/24 = 7.5^circ# is constructible so its cosine has an "exact" answer using square roots of positive numbers. Let's work it out.

#cos 15^circ = cos (45^circ - 30 ^circ) = cos 45 cos 30 + sin 45 sin 30 = sqrt{2}/2(sqrt{3}/2+1/2)=1/4(sqrt{6}+sqrt{2})#

Now the half angle formula; we choose the positive square root.

#cos 7.5^circ = sqrt{1/ 2 ( 1 + cos 15^circ ) } = sqrt{ 1/2(1 + 1/4(sqrt{6}+sqrt{2}))} = 1/2 sqrt(2 + sqrt(2 + sqrt(3))) #

#text{Area}=pi r^2 = pi( 1 + 32/ ( 1/2 sqrt(2 + sqrt(2 + sqrt(3))) )) #

#text{Area}= pi( 1 + 64/ ( sqrt(2 + sqrt(2 + sqrt(3))) )) #

Let's leave it there.

Aaliyah Anderson · Answer

undefined To find the area of the circle, we first need to find the radius. The chord's length is given as 8 units, and it subtends an angle of π/6 - π/8 radians.

The formula to find the radius (r) of the circle using the chord length (c) and the central angle (θ) subtended by the chord is:

$$r = \frac{c}{2\sin(\frac{\theta}{2})}$$

Substituting the given values, we have:

$$r = \frac{8}{2\sin(\frac{\pi/6 - \pi/8}{2})}$$

$$r = \frac{8}{2\sin(\frac{\pi}{48})}$$

$$r ≈ \frac{8}{2\sin(0.0654)}$$

$$r ≈ \frac{8}{2 \times 0.0654}$$

$$r ≈ \frac{8}{0.1308}$$

$$r ≈ 61.16$$

Now that we have the radius (r), we can use the formula for the area of a circle, which is:

$$A = πr^2$$

Substituting the value of the radius, we get:

$$A = π \times (61.16)^2$$

$$A ≈ π \times 3741.57$$

$$A ≈ 11753.04$$

Therefore, the area of the circle is approximately $11753.04$ square units.