The surface area of a square pyramid is 116 in.2 and the total area of the pyramid’s four triangular faces is 80 in.2
What is the length of one of the sides?
Let's start by using the formula for the surface area of a square pyramid:
SA = s^2 + 2s(l)
Where SA is the surface area, s is the length of one of the sides of the square base, and l is the slant height of the pyramid.
We are given that the surface area is 116 in.2, so we can write:
116 = s^2 + 2s(l)
We are also given that the total area of the four triangular faces is 80 in.2. Since each face is a triangle with base s and height l, we can write:
80 = 4(0.5s)(l)
80 = 2sl
Now we have two equations with two variables (s and l). We can use the second equation to solve for l in terms of s:
l = 40/s
Substitute this expression for l into the first equation:
116 = s^2 + 2s(l)
116 = s^2 + 2s(40/s)
116 = s^2 + 80
Simplify:
s^2 = 36
Take the square root of both sides:
s = 6
Therefore, the length of one of the sides is 6 inches.
To find the length of one side of the square base of the pyramid, we can use the formula for the surface area of a square pyramid which is:
Surface Area = Base Area + 4 * (0.5 * base * slant height)
Given that the surface area is 116 in^2 and the total area of the pyramid's four triangular faces is 80 in^2, we can set up the equation:
116 = Base Area + 4 * (0.5 * base * slant height)
We are given that the total area of the four triangular faces is 80 in^2, so:
80 = 4 * (0.5 * base * slant height)
Dividing both sides by 4 gives us:
20 = 0.5 * base * slant height
Now, let's find the value of the base area:
116 = Base Area + 20
Base Area = 116 - 20 = 96 in^2
Since the base of a square pyramid is a square, we know that the base area is equal to the length of one side squared. So:
base^2 = 96
To find the length of one side, we take the square root of both sides:
base = √96
Calculating the square root of 96, we get:
base ≈ 9.798
Therefore, the length of one side of the square base of the pyramid is approximately 9.798 inches.