A pyramid has a rectangular base with edges of length 10 and 24. The vertex of the pyramid is directly 13 units above the center of the base. What is the total surface area of the pyramid?

Question

A pyramid has a rectangular base with edges of length 10 and 24. The vertex of the pyramid is directly 13 units above the center of the base. What is the total surface area of the pyramid?

Answer 1

If the base is 10x24 then drawing a line connecting corners from lower left to upper right would be 26 inch in length (Pythagorean theorem). So looking again at the base, half the distance from corner to the center would be 13. If the height from the center to the top of pyramid is also 13, then there is a right isosceles triangle with lengths 13 on each side making the hypotenuse sqrt(338). This is then the length of the edge of the pyramid from the corner to the top. Now draw another isosceles triangle with base 10. The edges are sqrt(338) to you can determine the height by Pythagorean theorem to be sqrt(313). area of triangle is 1/2*B*h. Here b is 10, h = sqrt(313) and there are two of them....so the area of two smaller triangles is 10*sqrt(313). For the larger triangles, they also share same edge = length sqrt(338) but base is 24. So Height is sqrt(194), making those two triangle = 24*sqrt(194). Now add surface area of base = 240. Answer: 240+24*sqrt(194) + 10*sqrt(313)