At the base of the pyramid lies a right-angled triangle with a hypotenuse equal to C and an acute angle of 30 °.

At the base of the pyramid lies a right-angled triangle with a hypotenuse equal to C and an acute angle of 30 °. The side edges of the pyramid are inclined to the base plane at an angle of 45 °. Find the volume of the pyramid.

АС = с, ∠АСВ = 30 °, ∠ОАС = 45 °, the vertex point O has a projection at point K in the middle of the hypotenuse AC.

V = 1/3 * S main * h.

In triangle ABC:

S main = 1/2 * AB * BC,

AB = AC * sin30 ° = s * 1/2,

BC = AC * sin (90 ° – 30 °) = s * √3 / 2,

S main = 1/2 * AB * BC = 1/2 * s * 1/2 * s * √3 / 2.

At AOK:

AK = AC / 2 = s / 2,

∠AKO = 90 °,

∠ОАС = ∠АС = 45 °,

n = OK,

OK = AO * sin 45 ° = AO * 1 / √2,

AO = AK / sin 45 ° = s / 2 * √2,

h = OK = s / 2 * √2 * 1 / √2,

V = 1/3 * 1/2 * s * 1/2 * s * √3 / 2 * s / 2 * √2 * 1 / √2 = √3s3 / 48.