An equilateral triangle lies at the base of the prism, the side of which is 16. Find the lateral surface of the prism

| March 15, 2021 | education

An equilateral triangle lies at the base of the prism, the side of which is 16. Find the lateral surface of the prism if its lateral edge is 15, and its projection onto the base plane is one of the heights of the triangle.

In an equilateral triangle ABC, we determine the length of the height AH.

AH = AB * 3/2 = 16 * √3 / 2 = 8 * √3 cm.

The height АН, by condition, is the projection of the edge АА1. Then from the right-angled triangle АА1Н,

A1H ^ 2 = AA1 ^ 2 – AH ^ 2 = 225 – 192 = 33.

From the point H of the triangle ABC, we draw the height of the НK to the side of the AC. In a right-angled triangle AНK, the angle KAН = 30, then the length of the leg НK = AH / 2 = 8 * √4 / 2 = 4 * √3 cm.

From the right-angled triangle A1НK, we determine the hypotenuse A1K, which is the height of the side face of AA1C1C. A1K ^ 2 = A1H ^ 2 + HK ^ 2 = 33 + 48 = 81.

A1K = 9 cm.

Saa1c1c = Saa1b1v = AC * A1K = 15 * 9 = 135 cm2.

Svv1s1s = BC * BB1 = 16 * 15 = 240 cm2.

Then S side = 135 + 135 + 240 = 510 cm2.

Answer: The lateral surface area is 510 cm2.



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