The height of the cone is equal to the radius of the base. find the angle at the apex of the axial section of the cone.

The axial section of a cone is a plane passing through the axis.

The axial section of the cone is an isosceles triangle ∆ABS.

The height of an isosceles triangle divides it into two equal right angles. Consider one of them, ∆ABO.

Since the base radius is equal to the height, the angle between the height and the ∠ABO generatrix is ​​equal to the angle between the generatrix and the ∠BAO radius. Since the angle ∠AOB, between the height and the radius is 90 °, and the sum of all the angles of the triangle is 180 °, then:

∠ABO = ∠BAO = (180 ° – 90 °) / 2 = 90 ° / 2 = 45 °.

Thus, the angle at the vertex ∠В is equal to:

∠В = ∠ABO + ∠BAO;

∠В = 45 ° + 45 ° = 90 °.

Answer: the angle at the apex of the axial section of the cone is 90 °.