A metalworker is creating decorative points in the form of triangular pyranids to go on top of a fence she is building. She wants to know the surface area of each pyramid. Use the surface area formula to find the surface area of each pyramid based on the figure.
The formula for finding the surface area of a pyramid is:
Surface Area = 0.5 * base perimeter * slant height + base area
In this case, the base of the pyramid is a triangle with all sides equal in length, so it is an equilateral triangle. Let's assume that each side of the equilateral triangle base has a length of "s".
The formula for finding the area of an equilateral triangle is:
Area = (s^2 * sqrt(3)) / 4
Since all sides are equal, the base perimeter of the triangle is:
Base Perimeter = 3s
Now, let's assume the height of the pyramid (from the apex to the base) is "h".
The slant height of the pyramid can be found using the Pythagorean theorem, where the slant height ("l") is the hypotenuse of a right triangle with legs of h and s/2:
l^2 = (s/2)^2 + h^2
l^2 = s^2/4 + h^2
l = sqrt(s^2/4 + h^2)
Now, we can plug in the values into the surface area formula:
Surface Area = 0.5 * 3s * sqrt(s^2/4 + h^2) + (s^2 * sqrt(3)) / 4
Surface Area = 1.5s * sqrt(s^2/4 + h^2) + (s^2 * sqrt(3)) / 4
This is the formula you can use to find the surface area of each pyramid, given the side length "s" of the equilateral triangle base and the height "h" of the pyramid.