a circle having an area of 452 square inches is cut into segments by a chord, which is 6cm from the center of the circle. Find the area of the bigger segment.
π r² = 452 ... r = 12
the triangle formed by the chord and the radii from the ends has a central angle of 120º
so 2/3 of the circle (240º / 360º), plus the area of the triangle; is the area of the larger segment
To find the area of the bigger segment, we need to find the area of the entire circle and the area of the smaller segment.
1. Find the area of the entire circle:
The formula to find the area of a circle is A = πr^2, where "A" is the area and "r" is the radius. Since we know the area of the circle is 452 square inches, we can rearrange the formula to solve for the radius:
452 = πr^2
Divide both sides by π:
r^2 = 452/π
Take the square root of both sides to find the radius:
r = √(452/π)
2. Find the area of the smaller segment:
To find the area of a segment, we need to find the area of the sector formed by the chord and subtract the area of the triangle formed by the chord and two radii.
a. Area of the sector:
The angle of the sector formed by the chord can be found using trigonometry. The chord divides the circle into two equal angles, so we can use the inverse cosine function to find the angle:
cosθ = Adjacent/Hypotenuse
cosθ = 6/r
θ = acos(6/r)
The area of the sector is given by A_sector = (θ/360) * πr^2.
b. Area of the triangle:
The triangle formed by the chord and two radii is an isosceles triangle. We can find the area using the formula: A_triangle = (1/2) * b * h, where "b" is the base (chord length) and "h" is the height (distance from the center to the chord).
Since the chord is 6 cm from the center, the height can be found using the Pythagorean theorem:
h = sqrt(r^2 - 6^2)
The base length is simply the length of the chord, which is 6 cm.
Now we can calculate the area of the triangle.
3. Find the area of the bigger segment:
To find the area of the bigger segment, subtract the area of the smaller segment from the area of the entire circle.
Area of bigger segment = Area of circle - Area of smaller segment.
I hope this helps! Let me know if you have any further questions.