A metalworker is creating decorative points in the form of triangular pyramids to go on top of the 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.
To find the surface area of a pyramid, you need to calculate the area of each triangular face and add them together.
The formula for the surface area of a pyramid with a triangular base is:
Surface area = (base area) + (0.5 × perimeter of base × slant height)
In this case, the triangular base is the top of the fence, and each triangular face has the same measurements.
Let's say the base of the triangular pyramid has a length of "a" and a width of "b".
The base area is given by the formula:
Base area = 0.5 × (a × b)
The perimeter of the base is given by the formula:
Perimeter of base = a + b + sqrt(a^2 + b^2)
The slant height, which is the height of each triangular face, can be found using the Pythagorean theorem:
Slant height = sqrt(a^2 + b^2)
Therefore, the surface area of each pyramid is:
Surface area = (0.5 × (a × b)) + (0.5 × (a + b + sqrt(a^2 + b^2)) × sqrt(a^2 + b^2))