Express answer in exact form.
Find the area of the smaller segment whose chord is 8" long in a circle with an 8" radius
make a sketch and draw the sector.
Notice that if you take the segment away, you would be left with an equilateral triangle of sides 8, and of course the angle is 60°
The sector would be 1/6 of the area of the circle.
So find the area of the equilateral, and subtract that area from the area of the sector.
Let me know what you get.
Find the radius of the circle whose equation is (x^2 - 10x + 25) + (y^2 - 16y + 64) = 16.
To find the area of the smaller segment, we first need to find the angle of the sector formed by the chord.
Given that the chord length is 8", and the radius of the circle is also 8", we can use the formula for the chord length in terms of the radius and the angle of the sector:
Chord length = 2 * radius * sin(angle/2)
Rearranging the formula, we can solve for the angle:
sin(angle/2) = chord length / (2 * radius)
sin(angle/2) = 8 / (2 * 8)
sin(angle/2) = 1/2
To find the angle, we need to take the inverse sine (sin^(-1)) of both sides:
angle/2 = sin^(-1)(1/2)
angle/2 = 30°
Since the chord divides the circle into two equal segments, the total angle of the sector is 2 times the angle we just calculated:
angle = 2 * (angle/2)
angle = 2 * 30°
angle = 60°
Now, to find the area of the segment, we can use the formula:
Area of segment = (θ/360) * π * r^2
where θ is the angle of the sector and r is the radius of the circle.
Plugging in the values:
Area of segment = (60°/360°) * π * (8^2)
Area of segment = (1/6) * π * 64
Area of segment = (π * 64) / 6
Area of segment = (32π) / 3
So, the area of the smaller segment is (32π) / 3 square inches.