Find the area of the axial section of the cone if the radius of its base is 5m and the height is 7m.

A cone is a geometric body formed by rotating a right-angled triangle around its leg.

Since the axial section of the cone is an isosceles triangle, we will use Heron’s formula to calculate its area:

S = √p (p – a) (p – b) (p – c); where:

S is the area of ​​the triangle;

p – semi-perimeter (p = P / 2);

a, b – generator L;

c – diameter D.

To do this, you need to find the generator, which is the lateral side of this triangle.

Consider a triangle formed by the height, radius and generatrix of the cone. This triangle is rectangular. To calculate the length of the generator, we apply the Pythagorean theorem:

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

L ^ 2 = 7 ^ 2 + 5 ^ 2 = 49 + 25 = 74;

L = √74 = 8.6 m.

D = 2R;

D = 2 5 = 10 m.

p = (8.6 + 8.6 + 10) / 2 = 27.2 / 2 = 13.6 m.

S = √13.6 (13.6 – 8.6) (13.6 – 8.6) (13.6 – 10) = √13.6 5 5 3.6 = √1224 = 34.985 ≈ 35 m2.

Answer: the area of ​​the axial section of the cone is 35 m2.