Question

*These are regular polygons*
1.Find the area of a triangle with an apothem of 8 inches.
2.Find the area of a square with a radius of 12 inches.
3.Find the area of a pentagon with a hypotenuse of 10 inches.

Answer 1

1. To be speaking of an "apothem" of a triangle, the triangle must be equilateral.
The apothem is the distance from the centre (centroid) to a side, and the apothem must be 1/3 of the length of the median
so the median is 24 inches, which makes the height 24 inches.
then by knowing the ratio of sides of the 30-60-90 triangle, the side must be 48/√3
area = (1/2)(48/√3)(24) = 576/√3 square inches

2. We don't speak of a square having a radius.
Did you mean diameter?
if so let each side of square be x
x^2 + x^2 = 12^2
x = √98 =
area = x^2 = 98

3. Again, where does a hypotenuse come from in a pentagon?
Did you mean diagonal?
Find the internal angle. (you should get 108°)
From that you should be able to find all kinds of angles, depending on which lines you sketch.

Answer 2

a=8,r=16 , A=192*root Of 3.