The diagonal of the lateral grant of a regular triangular prism is 20cm, the side of the base is 12cm. Find its surface area.

1. To find the area of ​​the side surface S1, you need to find the height h of the side face of the prism.

2. By the Pythagorean theorem, we calculate h if, according to the condition of the problem, the diagonal of the lateral face of the rectangular shape is 20 cm, and the side of the base is 12 cm.

The diagonal of the side face divides it into two right-angled triangles, in which you can find h, which is the height of the prism.

h = (20 ^ 2 – 12 ^ 2) ^ 1/2 = (400 144) ^ 1/2 = 16 cm.

3. Let’s calculate S1 of the prism, which consists of the area of ​​three lateral faces.

S1 = 3 * 12 cm * 16 cm = 576 cm ^ 2.

4. Let’s calculate what is the area s of each of the two triangles lying at the base of the prima.

s = 1/2 base * height.

5. Calculate the height of the triangles at the base of the prism using the Pythagorean theorem.

Height = (side ^ 2 – 1/2 * side ^ 2) ^ 1/2 = (12 ^ 2 – 6 ^ 2) ^ 1/2 = 108 ^ 1/2 = 10.4 cm.

6. Find the area of ​​the base s.

s = 1/2 * 12 * 10.4 = 62.4 cm ^ 2.

7. Find the entire surface area of ​​the prism.

S = S1 + 2 * s = 576 + 2 * 62.4 = 700.8 cm ^ 2.

Answer: The surface area of ​​the prism is 700.8 square centimeters.