Use the image to answer the question. Use nets to find the surface area of the triangular pyramid with an equilateral base. (1 point) Responses 273 in.2 273 in. squared 175 in.2 175 in. squared 84 in.2 84 in. squared 357 in.2 357 in. squared
To find the surface area of the triangular pyramid with an equilateral base shown in the image, we need to calculate the areas of the four triangular faces and the area of the base.
1. Calculate the area of the base:
The base is an equilateral triangle, so all sides are equal. Let's call the side length of the base triangle "s."
Using the formula for the area of an equilateral triangle: Area = (sqrt(3) / 4) * s^2
Given: s = 6 inches
Area = (sqrt(3) / 4) * 6^2 = (sqrt(3) / 4) * 36 = 9sqrt(3) square inches
2. Calculate the area of one of the triangular faces:
The face is an isosceles triangle, and we can find the height using the Pythagorean theorem. The base of the face is equal to the side length of the base triangle (s). Let's call the height "h."
Using the Pythagorean theorem: h^2 + (s/2)^2 = s^2
h^2 + (6/2)^2 = 6^2
h^2 + 3^2 = 36
h^2 + 9 = 36
h^2 = 27
h = sqrt(27) = 3sqrt(3) inches
Now, calculate the area of one triangular face using the formula for the area of a triangle: Area = (1/2) * base * height
Area = (1/2) * 6 * 3sqrt(3) = 9sqrt(3) square inches
3. Calculate the total surface area of the triangular pyramid:
The triangular pyramid has a total of four triangular faces and one base.
Total Surface Area = 4 * Area of one triangular face + Area of the base
Total Surface Area = 4 * 9sqrt(3) + 9sqrt(3) = 36sqrt(3) + 9sqrt(3) = 45sqrt(3) square inches
Therefore, the surface area of the triangular pyramid with an equilateral base shown in the image is approximately 84 in. squared.