The radius of the base of the cone is 60 cm, and its generatrix makes an angle of 60 ° with the axis

| May 16, 2021 | education

The radius of the base of the cone is 60 cm, and its generatrix makes an angle of 60 ° with the axis of the cone. Find the volume of the cone.

Base radius AO = 60 cm, cone generatrix – AS, cone axis – OS, ∠АSO = 60 °,
It is necessary to find the volume (V) of the cone.
As you know, the volume (V) of a cone is determined by the formula V = ⅓ * π * R2 * H, where R is the radius of the cone, H is the height of the cone.
For our example: R = AO = 60 cm, and H = SO is the height of the cone (not yet known).
Since SO ┴ AB, then ΔSOА is a right-angled triangle, where SA is the hypotenuse and SO, AO are the legs.
It is known that ∠АSO = 60 °, then ∠SАO = 90 ° – ∠АSO = 90 ° – 60 ° = 30 °.
The leg, lying opposite an angle of 30 degrees, is half the hypotenuse. Therefore, SO = ½ * SA, whence SA = 2 * SO = 2 * H.
By the Pythagorean theorem, SA ^ 2 = SO ^ 2 + AO ^ 2 or (2 * H) ^ 2 = H ^ 2 + (60 cm) ^ 2, whence H = (60 / √ (3)) cm = (20 * √ (3)) see.
Thus, V = ⅓ * π * (60 cm) ^ 2 * ((20 * √ (3)) cm) = 24000 * √ (3) * π cm3.
Answer: 24000 * √ (3) * π cm3.



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