The plane passes through two generatrices of the cone and intersects the base of the cone along a chord

| July 28, 2021 | education

The plane passes through two generatrices of the cone and intersects the base of the cone along a chord, which is seen from the center of the base at an angle α. The section plane forms an angle β with the height of the cone. Find the area of the lateral surface of the cone if the height of the cone is H.

Let’s construct the radii OM and OK to the edges of the chord MK. Triangle MОК is isosceles, ОМ = AK = R, angle ОК = α.

OH is the height, bisector, and median of the IOC triangle.

In a right-angled triangle BOH, tgβ = OH / OB = OH / h.

OH = h * tgβ.

In a right-angled triangle MOH, the angle MOH = α / 2.

Cos (α / 2) = OH / OM = h * tgβ / R.

R = h * tanβ / Cos (α / 2).

In a right-angled triangle AOB, according to the Pythagorean theorem, AB ^ 2 = OB ^ 2 + R ^ 2 = h ^ 2 + (h * tanβ / Cos (α / 2)) ^ 2.

AB = h * √ (1 + (tanβ / Cos (α / 2)) ^ 2) see.

Side = π * R * AB = π * R * h * √ (1 + (tanβ / Cos (α / 2)) ^ 2) cm2.

Answer: Side = π * R * h * √ (1 + (tanβ / Cos (α / 2)) ^ 2) cm2.



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