Find the volume of the cone if its generatrix is 12 cm and the apex angle is 120 °.

| September 4, 2021 | education

To find the volume of a cone, you need to calculate its height and the radius of the base circle.

Since the length of the generatrix of the cone L = 12 cm and the apex angle α = 120 ° are known, we can calculate the value of the radius of the circle R and the height of the cone H.

Since L, H and R form a right-angled triangle, we can apply the following formulas:

1) Cos α / 2 = H / L.

2) Sin α / 2 = R / L.

Find the height from the 1st formula.

H = L * cos α / 2.

H = 12 cm * cos 60 °.

H = 12 cm * 1/2.

H = 6 cm.

Find the radius from the 2nd formula.

R = L * Sin α / 2.

R = 12 cm * Sin 60 °.

R = 12 cm * √3 / 2.

R = 6√3 cm.

Now we determine the volume of the cone V.

V = 1/3 * π * R2 * H.

V = 1/3 * π * (6√3) 2 * 6 cm3.

V = 216 π cm3.

Answer: the volume of the cone is 216 π cm3.



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