How do you calculate the surface area of a rectangular pyramid given only rectangle measures 5 X 3 slant height of 5 side is 4?

Answer 1

Consider the perpendicular height, h, as measured by a vertical line from the apex of the pyramid to the base. This meets the base at a distance of 1.5 from the 5 side and a distance of 2.5 from the 3 side.

Now, consider the triangle formed by the slant height on the 5 side the vertical height and the line from its bottom to the 5 side. This is a right angled triangle so that

h^2 + 1.5^2 = 4^2 and therefore h^2 = 12.25

Next consider the triangle formed by the slant height on the 3 side (s) the vertical height and the line from its bottom to the 3 side. This is also a right angled triangle so that

h^2 + 2.5^2 = s^2 and therefore s^2 = 12.25 - 2.5^2 = 6.

So the area of the base = 3*5 = 15 square units.

Area of each triangle on the 5 side = 1/2*5*4 = 10 square units

and area of each triangle on the 3 side = 1/2*3*6 = 9 square units.

So total area = 15 + 2*10 + 2*9 = 53 square units.