3.5 cm
3.9 cm
4.5 cm
A metalworker is creating decorative points in the form of triangular pyramids 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.
(1 point)
c m^ 2
To find the surface area of a triangular pyramid, we first need to find the area of the triangular base and the area of each of the three triangular faces.
Let's assume that the base of the pyramid is an equilateral triangle with side length 3.5 cm.
The formula to find the area of an equilateral triangle is:
Area = (sqrt(3)/4) * side length^2
Plugging in the values, we have:
Area = (sqrt(3)/4) * (3.5 cm)^2
≈ (0.866/4) * 12.25 cm^2
≈ 0.2165 * 12.25 cm^2
≈ 2.652 cm^2
Now, let's find the area of each of the three triangular faces.
We know that the height of the pyramid is given by the formula:
Height = sqrt(h^2 - (b/2)^2)
where h is the height of the triangular pyramid and b is the base of the triangular pyramid.
We are not given the height of the pyramid, but we can find it using the Pythagorean theorem.
In a right triangle with the height as the hypotenuse, one leg is h and the other leg is (b/2). The hypotenuse can be found using:
h = sqrt((b/2)^2 + h^2)
Rearranging the equation, we have:
h^2 = (b/2)^2 + h^2 - h^2
h^2 = (b/2)^2
h = b/2
Therefore, the height of the triangular pyramid is equal to half of its base.
Using this information, we can find the area of each triangular face using the formula:
Area = (base * height) / 2
Plugging in the values, we have:
Area = (3.5 cm * (3.5 cm/2) ) / 2
= (3.5 * 1.75) / 2
≈ 6.125 cm^2
Now, to find the surface area, we add the area of the base and the three triangular faces:
Surface Area = 2 * base area + 3 * triangular face area
= 2 * 2.652 cm^2 + 3 * 6.125 cm^2
≈ 5.304 cm^2 + 18.375 cm^2
≈ 23.679 cm^2
Therefore, the surface area of each pyramid is approximately 23.679 cm^2.