A right circular cone is made from of a radius of 5 cm, a central angle of 216 degrees, what is the volume of the cone in cubic cm?
The key here is that the circumference of the base of the cone is equal to the arc length of the original sector.
arc-length/10π = 216/360
arc-length = 6π
if radius of cone is r,
2πr = 6π
r = 3
by taking a cross-section of the cone, we get a right-angled triangle with hypotenuse of 5, height of h and base segment of 3
so h^2 + 3^2= 5^2
h = 4
vol of cone = (1/3)π(3^2)(4) = 12π cm^3
To find the volume of a right circular cone, you can use the formula:
Volume = (1/3) * π * r^2 * h
Where:
- 'r' is the radius of the base of the cone,
- 'h' is the height of the cone.
Since the problem statement does not provide the height of the cone, we need to find it first.
To do that, we can use the central angle given. In a circle, a central angle of 360 degrees corresponds to the entire circumference of the circle. Therefore, we can calculate the circumference of the base of the cone using the formula:
Circumference = 2 * π * r
Given the radius 'r' as 5 cm, we can calculate the circumference:
Circumference = 2 * π * 5 = 10π
Now, we can calculate the height 'h' of the cone using the central angle.
We know that the angle of 216 degrees represents a fraction of the entire circumference:
216 degrees / 360 degrees = Fraction of the circumference
Similarly, we can express the fraction of the circumference as a ratio with the total circumference:
Fraction of the circumference = Circumference of the base / Total circumference
Fraction of the circumference = 10π / 2π
Simplifying, we find:
Fraction of the circumference = 5
Thus, the height 'h' of the cone is equal to the radius 'r', which is 5 cm.
Now that we have the values of 'r' and 'h', we can substitute them into the volume formula:
Volume = (1/3) * π * r^2 * h
= (1/3) * π * (5^2) * 5
= (1/3) * π * 25 * 5
= (1/3) * π * 125
≈ 41.67π
Hence, the volume of the cone is approximately 41.67 times the value of π, in cubic cm.