finding semicircle area without radius

Question

In Algebra 2, I must find the area of a semi-circle without a stated radius. The problem is written as follows: The figure below (composed of a right triangle and a semi-circle) forms the base of a pyramid nine feet high. I must find the area of the base, then the volume of the pyramid. The bottom of the triangle is five feet. The left side of the triangle, which is slanted, is thirteen feet. The right side of the triangle, which is attached to the semi-circle, is a right angle. This procedure was explained in my textbook, but a radius for the semi-circle was always given. Is there a way for me to determine the radius based on the other sides? In the book, they usually used radii of two of three feet, if they were using this length unit. Should I write the steps for a radius of two or three?

Answer 1

James,

Absent a figure, your description is a little murky. What I have doped out is you have a right triangle whose side of unknown length is equal to the diameter of the semicircle, which is also equal to the base length of a square which serves as the bottom of a pyramid.  If so, then:

 

The right triangle can be solved given its hypotenuse (c) is 13 feet and the smallest side (a) is 5 ft long.  Using the c² = a² + b² formula, b² = 169 - 25 = 144.  So b = 12 feet.  And that is both the diameter of the circle and base length of the pyramid.

 

Hence, area of semicircle = 0.5 Π (6)² = 18 Π ft² .

           area of pyramid's square base  = (12)² = 144 ft² .

           volume of 9 ft tall pyramid = (1/3)(base area)(height) = (1/3)(144)(9) = 432 ft³ .