Find the perimeter and area of a segment of a circle of radius 8cm, cut off by a chord that subtended a central angle of:. a 120o. b 3/4
a) To find the perimeter and area of a segment of a circle, we need to know the length of the chord and the height of the segment.
We are given that the radius is 8cm.
Using the central angle of 120o, we can divide the circle into two equal parts, each containing an angle of 60o. The chord that subtends this angle will bisect the circle into two equal parts. This chord is also the diameter of the circle, and its length can be found using the formula:
diameter = 2 x radius = 2 x 8cm = 16cm
To find the length of the chord that subtends the central angle of 120o, we can use the following formula:
length of chord = 2 x radius x sin(angle/2)
where angle is the central angle in degrees.
Substituting in the values, we get:
length of chord = 2 x 8cm x sin(60o) ≈ 13.86cm
Now, we need to find the height of the segment, which is the distance between the chord and the center of the circle.
We can divide the segment into two right triangles, with the chord as their hypotenuse. The height is the perpendicular drawn from the center of the circle to the chord, bisecting it.
Using one of the triangles, we can find the height as:
height = radius x cos(angle/2)
Substituting in the values, we get:
height = 8cm x cos(60o/2) = 8cm x cos(30o) ≈ 6.93cm
Now, we can calculate the perimeter and area of the segment using the following formulas:
perimeter = length of chord + 2 x height
area = (angle/360) x π x radius2 - 0.5 x length of chord x height
Substituting in the values, we get:
perimeter = 13.86cm + 2 x 6.93cm ≈ 27.72cm
area = (120/360) x π x 82 - 0.5 x 13.86cm x 6.93cm ≈ 88.35cm2
Therefore, the perimeter of the segment is approximately 27.72cm and its area is approximately 88.35cm2.
b) We are given that the central angle is 3/4 of the total angle, which means it is 270o.
Using the same method as above, we can find the length of the chord as:
length of chord = 2 x 8cm x sin(270/2) ≈ 15.45cm
The height of the segment can be found as:
height = 8cm x cos(270/2) = 8cm x cos(135o) ≈ 2.83cm
Using the formulas for perimeter and area of a segment, we get:
perimeter = 15.45cm + 2 x 2.83cm ≈ 21.11cm
area = (270/360) x π x 82 - 0.5 x 15.45cm x 2.83cm ≈ 39.27cm2
Therefore, the perimeter of the segment is approximately 21.11cm and its area is approximately 39.27cm2.
To find the perimeter and area of a segment of a circle, we need to know the radius and the central angle subtended by the chord. In this case, the radius is given as 8 cm.
a) For a central angle of 120 degrees:
To find the perimeter of the segment, we need to find the length of the arc of the circle that corresponds to the central angle of 120 degrees. The formula to find the length of an arc is:
Length of Arc = (angle in degrees / 360) * Circumference of the Circle
Circumference of the Circle = 2 * pi * radius
Circumference of the Circle = 2 * 3.14 * 8 cm
Circumference of the Circle = 50.24 cm
Length of Arc = (120 / 360) * 50.24 cm
Length of Arc = (1/3) * 50.24 cm
Length of Arc = 16.75 cm
The perimeter of the segment is the sum of the arc length and the lengths of the two radii:
Perimeter of the Segment = Arc Length + 2 * Radius
Perimeter of the Segment = 16.75 cm + 2 * 8 cm
Perimeter of the Segment = 16.75 cm + 16 cm
Perimeter of the Segment = 32.75 cm
To find the area of the segment, we need to find the area of the sector formed by the central angle and subtract the area of the triangle formed by the radii and the chord. The formula to find the area of a sector is:
Area of Sector = (angle in degrees / 360) * pi * (radius^2)
Area of Sector = (120 / 360) * 3.14 * (8 cm)^2
Area of Sector = (1/3) * 3.14 * 64 cm^2
Area of Sector = 67.03 cm^2
The triangle formed by the radii and the chord is an equilateral triangle, since all sides are equal to the radius. The formula to find the area of an equilateral triangle is:
Area of Triangle = (sqrt(3) / 4) * (side^2)
Area of Triangle = (sqrt(3) / 4) * (8 cm)^2
Area of Triangle = (sqrt(3) / 4) * 64 cm^2
Area of Triangle = 11.08 cm^2
Therefore, the area of the segment is the difference between the area of the sector and the area of the triangle:
Area of Segment = Area of Sector - Area of Triangle
Area of Segment = 67.03 cm^2 - 11.08 cm^2
Area of Segment = 55.95 cm^2
b) For a central angle of 3/4:
To find the perimeter of the segment, we follow the same steps as in part a, but using the central angle of 3/4 instead.
Length of Arc = (angle in degrees / 360) * Circumference of the Circle
Length of Arc = (3/4) * 50.24 cm
Length of Arc = 37.68 cm
Perimeter of the Segment = Arc Length + 2 * Radius
Perimeter of the Segment = 37.68 cm + 2 * 8 cm
Perimeter of the Segment = 37.68 cm + 16 cm
Perimeter of the Segment = 53.68 cm
To find the area of the segment, we follow the same steps as in part a, but using the central angle of 3/4 instead.
Area of Sector = (angle in degrees / 360) * pi * (radius^2)
Area of Sector = (3/4) * 3.14 * (8 cm)^2
Area of Sector = (3/4) * 3.14 * 64 cm^2
Area of Sector = 150.72 cm^2
Area of Triangle = (sqrt(3) / 4) * (side^2)
Area of Triangle = (sqrt(3) / 4) * (8 cm)^2
Area of Triangle = (sqrt(3) / 4) * 64 cm^2
Area of Triangle = 11.08 cm^2
Area of Segment = Area of Sector - Area of Triangle
Area of Segment = 150.72 cm^2 - 11.08 cm^2
Area of Segment = 139.64 cm^2