The base of a right-angled prism is a right-angled triangle with a 7cm leg and a hypotenuse √53.

The base of a right-angled prism is a right-angled triangle with a 7cm leg and a hypotenuse √53. The smallest side face of the prism is a square. Find the volume and area of the lateral surface of the prism.

Knowing the hypotenuse and one of the legs of the right triangle, we can find the second leg:

b ^ 2 = c ^ 2 – a ^ 2 = (√53) ^ 2 – 7 ^ 2 = 53 – 49 = 4 = 22;

b = 2 cm.

Obviously, the smallest side face of the prism is the face, one of the sides of which is the smallest side of the base. If the smallest side face is a square, then the height of the prism is equal to the smaller side of the base:

h = b = 2 cm.

The base area is equal to half the product of the legs:

Sb = 0.5 * 7 * 2 = 7 cm2.

The volume of the prism is equal to the product of the base area by the height:

V = Sbn * h = 7 * 2 = 14 cm3.

The lateral surface area is equal to the product of the base perimeter by the height:

Sside = h * P = 2 * (7 + 2 + √53) = 18 + 2√53 ≈ 32.56 cm2.