The height of the cone is 6cm, the angle at the apex of the axial section is 90 degrees.

The height of the cone is 6cm, the angle at the apex of the axial section is 90 degrees. Find the area of the lateral surface of the cone.

In order to calculate the area of ​​the lateral surface of the cone, you must use the formula:

S = πrl, where:

r is the radius of the base;

l – generator.

Consider the axial section of the cone. It has the shape of an isosceles triangle. Let’s designate it ABC. BO is the height of the cone, as well as the axial section.

To calculate the length of the radius and the generator, take, for example, the triangle ΔАВН.

Since the height of an isosceles triangle is also a bisector, then:

∠АВН = ∠В / 2;

∠AВН = 90º / 2 = 45º

Since in a triangle the sum of all angles is 180º, then:

∠ВAН = 180º – ∠AВН – ∠ВНА;

∠ВAН = 180º – 90º – 45º = 45º.

Based on this, we see that this triangle is also isosceles, in which:

AH = BH = 6 cm.

We find the generator AB behind the Pythagorean theorem:

AB ^ 2 = BH ^ 2 + AH ^ 2;

AB ^ 2 = 6 ^ 2 + 6 ^ 2 = 36 + 36 = 72;

AB = √72 ≈ 8.5 cm.

S = 3.14 * 6 * 8.5 = 160.14 cm2.

Answer: the area of ​​the lateral surface of the cone is 160.14 cm2.