The height of the cone is equal to the radius R of its base. A plane is drawn through the apex of the cone, cutting

| April 11, 2021 | education

The height of the cone is equal to the radius R of its base. A plane is drawn through the apex of the cone, cutting off an arc of 60 degrees from the base circumference. Find S section

From the right-angled triangle AOB, by the Pythagorean theorem, we determine the length of the hypotenuse AB.

AB ^ 2 = OA ^ 2 + OB ^ 2 = 2 * R ^ 2.

AB = R * √2 cm.

Then ВK = BM = AB = R * √2 cm.

Since the chord KM cuts off an arc of 60, the triangle OKM is equilateral, OK = OM = KM = R cm.

The ВKM triangle is isosceles. Let’s build in it the height of the VN, which is also its median, then KН = KM / 2 = R / 2 cm.

Then, by the Pythagorean theorem, in the triangle ВKН, BH ^ 2 = BK ^ 2 – KH ^ 2 = 2 * R ^ 2 – R ^ 2/4 = 7 * R ^ 2/4.

BH = √7 * R / 2 cm.

Determine the cross-sectional area.

Ssection = KM * ВН / 2 = R * (√7 * R / 2) / 2 = R^2 * √7 / 4 cm2.

Answer: The cross-sectional area is R^2 * √7 / 4 cm2.



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