A plane is drawn through the apex of the cone, cutting off a quarter of its base circle.

| March 16, 2021 | education

A plane is drawn through the apex of the cone, cutting off a quarter of its base circle. Find the total surface area of the cone if the base radius is R and the section angle at the top of the cone is 60 degrees.

Since the section cuts off the fourth part of the circle at the base of the cone, the central angle

OKM = 360/4 = 90.

Then, in the right-angled triangle ОKМ, by the Pythagorean theorem, we define the length of the chord KМ.

KM ^ 2 = OM ^ 2 + OK ^ 2 = 2 * R ^ 2.

KС = R * √2 cm.

The KВM triangle is isosceles, since ВK = ВM as generators of the cone, and the angle at the apex of the section, by condition, is 60, then the KВM triangle is equilateral ВK = ВM = KM = R * √2 cm.

Determine the area of ​​the base of the cone.

Sb = n * R2 cm2.

Let us determine the area of ​​the lateral surface of the cone.

Sside = n * R * L = n * R * R * √2 = n * R ^ 2 * √2 cm2.

Then Spol = S main + S side = n * R ^ 2 * + n * R ^ 2 * √2 = n * R ^ 2 * (1 + √2) cm2.

Answer: The area of ​​the cone is equal to n * R ^ 2 * (1 + √2) cm2.



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