Question

a paper cone has a base diameter of 8 cm and a height of 3 cm
a. calculate the volume of the cone
b. calculate the curved surface area of the cone
c. if the cone is cut and opened out into the sector of a circle, what is the angle of the sector?

Answer 1

a. To calculate the volume of the cone, we use the formula V = 1/3 * π * r^2 * h, where r is the radius of the base and h is the height of the cone.

Since the base diameter is 8 cm, the radius (r) is half of that, which is 4 cm. So, r = 4 cm and h = 3 cm.

V = 1/3 * π * 4^2 * 3
V = 1/3 * π * 16 * 3
V = 16π cm^3

Therefore, the volume of the cone is 16π cubic centimeters.

b. To calculate the curved surface area of the cone, we use the formula A = π*r*l, where l is the slant height of the cone.

The slant height can be calculated using the Pythagorean theorem: l = √(r^2 + h^2)
l = √(4^2 + 3^2)
l = √(16 + 9)
l = √25
l = 5 cm

A = π * 4 * 5
A = 20π cm^2

Therefore, the curved surface area of the cone is 20π square centimeters.

c. To find the angle of the sector when the cone is cut and opened out into a sector of a circle, we need to find the circumference of the base of the cone and then determine the full circumference of the circle.

The circumference of the base is C = 2 * π * r
C = 2 * π * 4
C = 8π

The full circumference of the circle is 2πr, where r is the radius of the base (4 cm).

Thus, the angle of the sector of the circle is calculated as:
angle = 360 * (C / (2 * π * r))
angle = 360 * (8π / 8π)
angle = 360 degrees

Therefore, the angle of the sector when the cone is cut and opened out into a sector of the circle is 360 degrees.