# The side of the base of a regular triangular prism is 4 m, the diagonal of the side face is 5 m

**The side of the base of a regular triangular prism is 4 m, the diagonal of the side face is 5 m. Find the area of the side surface of the prism?**

ABCA1B1C1 – regular triangular prism. The area of the lateral surface of the prism is:

Sside = Posn * H,

where Rosn is the perimeter of the base, H is the length of the lateral edge.

1. Since the ABCA1B1C1 prism is regular, a regular triangle lies at its base. The perimeter of the right triangle is:

P = 3a,

where a is the side of the triangle.

Psn = 3 * 4 = 12 (m).

2. Since the prism ABCA1B1C1 is correct, all its side faces are equal to each other. Consider the side face of AA1C1C: AA1C1C is a rectangle, AC1 = 5 m is the diagonal, AC = A1C1 = 4 m is the length of the rectangle, AA1 = CC1 is the width of the rectangle. Consider a right-angled triangle AC1C: AC1 = 5 m – hypotenuse, AC = 4 m and CC1 – legs. By the Pythagorean theorem:

CC1 = √ (AC1 ^ 2 – AC ^ 2);

CC1 = √ (5 ^ 2 – 4 ^ 2) = √ (25 – 16) = √9 = 3 (m).

3. The area of the lateral surface of the prism ABCA1B1C1 is equal to:

Side = 12 * 3 = 36 (m ^ 2).

Answer: Side = 36 m ^ 2.