The height of the cone is 10cm. Find the area of the section passing through the apex of the cone and the chord

The height of the cone is 10cm. Find the area of the section passing through the apex of the cone and the chord of the base, contracting an arc of 60 degrees, if the plane of the section forms 30 degrees with the plane of the base of the cone.

Find the radius of the base.
Consider the triangle ΔABO, formed by the radius, height and secant plane ABC. This triangle is rectangular.

Since one of its legs and the value of the opposite angle to it are known, we will use the tangent of this angle to calculate the second leg:

tg A = BO / AO;

AO = BO / tg A;

tg 30º = 0.5774;

AO = 10 / 0.5774 ≈ 17.3 cm.

This leg is the radius of the base.

Let’s find the chord AC.
l = 2r sin (α / 2);

sin (60º / 2) = sin 30º = 1/2 = 0.5;

AC = 2 * 17.3 * 0.5 = 17.3 cm.

Let us calculate the generatrix of the cone.
Consider the axial section. The triangle formed by the height, radius of the base and the generatrix is ​​rectangular, therefore, to calculate the length of the generatrix, we apply the Pythagorean theorem:

L ^ 2 = r ^ 2 + h ^ 2

L ^ 2 = 17.3 ^ 2 + 10 ^ 2 = 299.29 + 100 = 399.29;

L = √399.29 ≈ 20 cm.

Now we will find the sectional area ABC. Since it has the shape of a triangle, we apply Heron’s formula:
S = √p (p – a) (p – b) (p – c);

p = (a + b + c) / 2;

p = (20 + 20 + 17.3) / 2 = 57.3 / 2 = 28.65 cm;

S = √28.65 (28.65 – 20) (28.65 – 20) (28.65 – 17.3) = √28.65 * 8.65 * 8.65 * 11 , 35 = √24330.59 ≈ 156 cm2.

Answer: the cross-sectional area of ​​the cone is 156 cm2.