SOLUTION: A square prism in inscribed in a cylinder. The prism has a height of 12 inches and the cylinder has a radius of 5 inches. What's the lateral area and surface area of the prism and

Algebra ->  Surface-area -> SOLUTION: A square prism in inscribed in a cylinder. The prism has a height of 12 inches and the cylinder has a radius of 5 inches. What's the lateral area and surface area of the prism and       Log On


   

Question 622471: A square prism in inscribed in a cylinder. The prism has a height of 12 inches and the cylinder has a radius of 5 inches. What's the lateral area and surface area of the prism and the cylinder
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I assume that from above you just see the bases of the prism and the cylinder, looking like a square inscribed in a circle, like this:
The diameter of the circle is 10 inches, the length of the diagonal of the square.
Two sides of the square, of length x inches, form a right triangle with the diagonal for a hypotenuse, so
x%5E2%2Bx%5E2=100 --> 2x%5E2=100 --> x%5E2=50 --> x=sqrt%2850%29=sqrt%2825%2A2%29=sqrt%2825%29%2Asqrt%282%29=5sqrt%282%29
The area of the circle (with radius 5) is pi%2A5%5E2=25pi square inches.
That circle is one of the two congruent (equal) bases of the cylinder.
The circumference is 2%2Api%2A5=10pi inches, and the lateral area of the cylinder is that times the height, or
10pi%2A12=120pi square inches.
So the total surface area of the cylinder is
120pi%2B2%2A25pi=170pi square inches.
The area of the square is x%5E2=50 square inches.
That square is one of the two congruent (equal) bases of the prism.
The perimeter of the square is 4%2A5sqrt%282%29=20sqrt%282%29 inches, and the lateral area of the prism is that times the height, or
20sqrt%282%29%2A12=120sqrt%282%29 square inches.
So the total surface area of the cylinder is
120sqrt%282%29%2B2%2A50=120sqrt%282%29%2B100 square inches.