Find the ratio of the area of the lateral surface of the cone to the area of the base if the angle between

Find the ratio of the area of the lateral surface of the cone to the area of the base if the angle between the height of the cone about the generatrix is 45 degrees

Let us write down the formulas for the area of the base and the lateral surface of the cone.
S main. = π * r²;
S side. = π * r * l, (l is the generator of the cone).
By the condition of the problem, the angle between the generatrix of the cone and the height is 45 °, which means that we have a right-angled isosceles triangle in which the legs are equal (r = h). By the Pythagorean theorem, we find the generator – the hypotenuse.
l = √ (r² + r²) = √2r² = r√2.
Find the lateral surface area:
S side. = π * r * l = π * r * r√2 = π * r²√2.
Find the area ratio:
S side. / S main. = π * r²√2 / π * r² = √2.
Answer: the area ratio is equal to √2.