Find the generatrix of the cone, the height of which is 12 cm, and the generatrix is inclined to the base

Find the generatrix of the cone, the height of which is 12 cm, and the generatrix is inclined to the base plane at an angle of 45 °.

Consider a right-angled triangle formed by the height of the cone, the radius of its base and the generatrix. Height – leg, opposite to the angle between the generatrix and the base plane, the generatrix – hypotenuse. The ratio of the opposite leg to the hypotenuse is the sine of the angle. Means:

sin 45 ° = h / L.

Hence:

L = h / sin 45 ° = 12 / (√2 / 2) = 12√2 ≈ 16.97 cm.