Suppose you have an equilateral triangle. The area of the triangle is exactly 1200 square centimetres. Now suppose you have twenty of those triangles. It's possible to assemble those twenty triangles into a closed three-dimensional shape, a regular polyhedron.
What would be the volume, in cubic centimetres, of the largest sphere that could fit inside the shape?
Please stop cheating at neopets!
The formula that you need for the radius of the inscribed sphere can be found in the link below, but deriving it would be rather difficult:
http://en.wikipedia.org/wiki/Icosahedron
Once you have the radius, compute the sphere volume in the usual way.
The Wikipedia reference gives the sphere radius in terms of the triangle side length, a. In your case, the side length is
a = [4/sqrt3 * A]^1/2 = 52.643 cm
To find the volume of the largest sphere that could fit inside the regular polyhedron created by assembling twenty equilateral triangles, you can follow these steps:
Step 1: Find the side length of each equilateral triangle.
Since the area of one of the triangles is given as 1200 square centimeters, we can use the formula for the area of an equilateral triangle: Area = (sqrt(3) / 4) * side^2, where side is the length of one side of the triangle.
1200 = (sqrt(3) / 4) * side^2
Multiplying both sides by 4 / sqrt(3):
(4 * 1200) / sqrt(3) = side^2
4800 / sqrt(3) = side^2
Taking the square root of both sides:
side = sqrt(4800 / sqrt(3))
Step 2: Find the height of one equilateral triangle.
The height of an equilateral triangle can be found using the formula: Height = (sqrt(3) / 2) * side.
height = (sqrt(3) / 2) * sqrt(4800 / sqrt(3))
height = sqrt(3 * 4800) / 2
Step 3: Find the volume of the regular polyhedron.
To find the volume of the regular polyhedron, you can multiply the base area of one equilateral triangle by the height of one triangle and multiply that by the number of triangles in the polyhedron (twenty in this case).
base area = (sqrt(3) / 4) * side^2
volume = (base area) * (height) * (number of triangles)
volume = (sqrt(3) / 4) * (sqrt(4800 / sqrt(3))^2) * (sqrt(3 * 4800) / 2) * 20
Simplifying the expression gives the volume of the regular polyhedron.
Once you have the volume of the regular polyhedron, the largest sphere that can fit inside it would have a volume equal to 4/3 of the volume of the polyhedron.
You can calculate the volume of the largest sphere by multiplying the volume of the polyhedron by 4/3.