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

**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.