At the base of the pyramid lies a rectangle. The height of the pyramid is h.

At the base of the pyramid lies a rectangle. The height of the pyramid is h. Find the volume of a pyramid if it is known that all five of its faces are of the same size.

1. Apply the Pythagorean theorem to the right-angled triangles SOM and SON:

SM ^ 2 = h ^ 2 + (a / 2) ^ 2;
SN ^ 2 = h ^ 2 + (b / 2) ^ 2.
2. From the equality of the areas of the SAD and SCD faces it follows:

SM * CD = SN * AD;
SM * b = SN * a;
SM ^ 2 * b ^ 2 = SN ^ 2 * a ^ 2;
(h ^ 2 + (a / 2) ^ 2) ^ 2 * b ^ 2 = (h ^ 2 + (b / 2) ^ 2) ^ 2 * a ^ 2;
h ^ 2b ^ 2 + a ^ 2 * b ^ 2/4 = h ^ 2a ^ 2 + b ^ 2 * a ^ 2/4;
h ^ 2b ^ 2 = h ^ 2a ^ 2;
b ^ 2 = a ^ 2;
b = a.
The base of the pyramid is a square.

3. From the equality of the areas of the base and the faces, we get:

1/2 * SN * a = a ^ 2;
SN = 2a;
SN ^ 2 = 4a ^ 2;
h ^ 2 + (a / 2) ^ 2 = 4a ^ 2;
4h ^ 2 + a ^ 2 = 16a ^ 2;
15a ^ 2 = 4h ^ 2;
a ^ 2 = 4h ^ 2/15.
4. The volume of the pyramid:

V = 1/3 * h * S;
V = 1/3 * h * a ^ 2;
V = 1/3 * h * 4h ^ 2/15 = 4h ^ 3/45.
Answer: 4h ^ 3/45.