Challenge. Difficulty level - medium

[ranga519]

In a regular quadrangular pyramid the area of a side surface is twice as much as
the area of the base. Find the ratio of the height to the length of the side of the base
of the pyramid.

 

[jvicens]

I would approach this problem using mathematical principles and equations. Let's assume that the base of the pyramid is a square with side length x and the height of the pyramid is h.

First, we can calculate the area of the base by using the formula for the area of a square, which is A = x^2. Since the area of the side surface is twice the area of the base, we can write this as 2A = 2x^2.

Next, we can use the formula for the surface area of a regular quadrangular pyramid, which is SA = (1/2)pl, where p is the perimeter of the base and l is the slant height. Since the base is a square, the perimeter is 4x. And since we know that the area of the side surface is twice the area of the base, we can write this as 2x^2 = (1/2)(4x)(l).

Simplifying this equation, we get 2x^2 = 2xl. Dividing both sides by 2x, we get x = l. This means that the slant height is equal to the side length of the base.

Finally, we can use the Pythagorean theorem to find the height of the pyramid. The height, h, is the hypotenuse of a right triangle with one leg being x and the other leg being l. So, using the Pythagorean theorem, we can write h^2 = x^2 + l^2. Since we know that x = l, we can substitute this into the equation to get h^2 = 2x^2. Taking the square root of both sides, we get h = √2x.

Therefore, the ratio of the height to the length of the side of the base is √2 : 1. In other words, the height is approximately 1.414 times the length of the side of the base.