Find the volume of the straight prism ABCA1B1C1 in which the angle ACB = 90 degrees.

Find the volume of the straight prism ABCA1B1C1 in which the angle ACB = 90 degrees. AB = BB1 = a. AC = CB.

The straight prism is a prism, at the base of which there is a triangle ABC with the same legs AC and CB. The hypotenuse of the triangle AB = a. The height of the prism is h = a. To find the volume, it remains for us to find the area of the base.

Let us denote the length of the legs by x.

Then the equality AC ^ 2 + BC ^ 2 = AB ^ 2 can be written in terms of a given value of the length of the hypotenuse a:

x ^ 2 + x ^ 2 = a ^ 2;

2x ^ 2 = a ^ 2;

x ^ 2 = a ^ 2/2

x = a / √2.

The area of a right-angled triangle S is equal to half the product of the legs:

S = (a / √2 * a / √2) / 2 = a ^ 2/4.

Prism volume:

V = Sh = a ^ 2/4 * a = a ^ 3/4.

Answer: V = a ^ 3/4.