Find the area of a segment formed by a chord 8" long in a circle with radius of 8".
as we all know, the side of an inscribed hexagon is equal to the radius.
So, the angle subtending the side is π/3
a = 1/2 r^2 (θ - sinθ)
= 1/2 * 64 (π/3 - √3/2)
= 32/6 (2π - 3√3)
More generally, the angle θ which subtends a chord of length s in a circle of radius r is
sin θ/2 = (s/2)/r = s/2r
In this problem, s=r=8, so
sin θ/2 = 1/2
θ = π/3
and away we go.
To find the area of a segment formed by a chord in a circle, we need to use the formula:
Area of Segment = Area of Sector - Area of Triangle
1. Firstly, let's find the measure of the central angle formed by the chord. Since the chord is 8" long and the radius is 8", the central angle can be found using the formula:
Central angle = 2 * arcsin(chord length / (2 * radius))
= 2 * arcsin(8 / (2 * 8))
= 2 * arcsin(1)
= 2 * 90°
= 180°
2. Now, let's find the area of the sector. The formula for the area of a sector is:
Area of Sector = (central angle / 360°) * π * radius^2
= (180° / 360°) * π * 8^2
= 0.5 * π * 64
= 32π square inches
3. Next, let's find the area of the triangle formed by the chord. To do this, we need to find the height of the triangle by using the formula:
Height of Triangle = radius - (chord length / 2)
= 8 - (8 / 2)
= 8 - 4
= 4 inches
The area of the triangle can be found using the formula:
Area of Triangle = (chord length / 2) * height
= (8 / 2) * 4
= 4 * 4
= 16 square inches
4. Finally, we can find the area of the segment by subtracting the area of the triangle from the area of the sector:
Area of Segment = Area of Sector - Area of Triangle
= 32π - 16
≈ 86.389 square inches
Therefore, the area of the segment formed by a chord 8" long in a circle with a radius of 8" is approximately 86.389 square inches.
To find the area of a segment formed by a chord in a circle, we need to know the length of the chord and the radius of the circle.
Given:
Chord length (c) = 8 inches
Radius (r) = 8 inches
To find the area of the segment, we can follow these steps:
1. Find the central angle (θ) of the segment.
The central angle can be calculated using the formula:
θ = 2 * arcsin(c / (2 * r))
Here, arcsin refers to the inverse sine function.
Substituting the values:
θ = 2 * arcsin(8 / (2 * 8))
= 2 * arcsin(1)
= 2 * (pi / 2)
= pi radians
2. Find the area of the sector (A_sector) formed by the central angle.
The area of a sector can be calculated using the formula:
A_sector = (θ / 2) * r^2
Substituting the values:
A_sector = (pi / 2) * (8^2)
= 16pi square inches
3. Find the area of the triangle (A_triangle) formed by the chord and its corresponding arc.
The area of a triangle can be calculated using the formula:
A_triangle = (1/2) * c * r
Substituting the values:
A_triangle = (1/2) * 8 * 8
= 32 square inches
4. Subtract the area of the triangle from the area of the sector to find the area of the segment.
A_segment = A_sector - A_triangle
= 16pi - 32
= 16(pi - 2) square inches
Therefore, the area of the segment formed by the chord in a circle with a radius of 8 inches is 16(pi - 2) square inches.