The height of the cone is 20 and the radius of its base is 25. find the area of the section

The height of the cone is 20 and the radius of its base is 25. find the area of the section through the vertex if the distance from the center of the base of the cone is 12.

In order to find the cross-sectional area ABC, it is necessary to find the length of the generatrix, which is equal to the length of the segments AB and BC, as well as the segment AC, which is the base of the triangle ABC.

To calculate the length of the generatrix, consider the axial section of this cone. The triangle formed by the height, radius and generatrix is ​​rectangular. Therefore, to calculate the generator, we apply the Pythagorean theorem:

L ^ 2 = r ^ 2 + h ^ 2;

L ^ 2 = 25 ^ 2 + 20 ^ 2 = 625 + 400 = 1025;

L = √1025 ≈ 32 cm.

Consider the base of the cone. The triangle ΔAO is isosceles, in which AO and OC are the lateral sides, AС is the base, OH is the height.

ΔAOH – rectangular.

To calculate AN, we apply the Pythagorean theorem:

AO ^ 2 = OH ^ 2 + AH ^ 2;

AH ^ 2 = AO ^ 2 – OH ^ 2;

AH ^ 2 = 25 ^ 2 – 12 ^ 2 = 625 – 144 = 481;

AH = √481 ≈ 21.9 cm.

HC = AH = 21.9 cm.

AC = AH + HC;

AC = 21.9 + 21.9 = 43.8 cm.

Since this section has the shape of a triangle, we will use Heron’s formula to calculate its area:

S = √p (p – a) (p – b) (p – c);

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

p = (43.8 + 32 + 32) / 2 = 107.8 / 2 = 53.9 cm;

S = √53.9 * (53.9 – 43.8) * (53.9 – 32) * (53.9 – 32) = √53.9 * 10.1 * 21.9 * 21.9 = √261094.888 ≈ 511 cm2.

Answer: the cross-sectional area of ​​the cone is 511 cm2.