Question

The surface area of the pyramid. The side lengths of the base are equal.

Triangular pyramid with base edge of 8 inches, and a slant height measuring 14 inches, on a triangular face. Apothem goes from the vertex to the midpoint of a base edge and measures 6.9 inches.

Answer 1

the surface area of the triangular pyramid is 227.6 square inches.

Explanation:

To find the surface area of the triangular pyramid, we need to calculate the area of the triangular base and the area of each triangular face, and then add them together.

The area of the triangular base can be found using the formula:

Base Area = (1/2) x Base Edge Length x Apothem

Substituting the given values, we get:

Base Area = (1/2) x 8 x 6.9 = 27.6 square inches

To find the area of each triangular face, we can use the formula:

Face Area = (1/2) x Base Edge Length x Slant Height

Substituting the given values, we get:

Face Area = (1/2) x 8 x 14 = 56 square inches

Since the triangular pyramid has 4 faces, we need to multiply the area of each face by 4 to get the total surface area. Therefore, the surface area of the triangular pyramid is:

Total Surface Area = 4 x Face Area + Base Area

= 4 x 56 + 27.6

= 227.6 square inches

So the surface area of the triangular pyramid is 227.6 square inches.

Answer 2

To find the surface area of the pyramid, we need to find the area of each of the four triangular faces and add them up.

First, let's find the length of the altitude of the triangle formed by the base edge and apothem. We can use the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the legs of the right triangle and c is the hypotenuse (slant height).

a = 4 (half of the base edge)

b = 6.9 (apothem)

c = 14 (slant height)

Substituting values:

4^2 + 6.9^2 = 14^2
16 + 47.61 = 196
63.61 = 196 - 16
63.61 = 180.00
√63.61 = 7.98

So the altitude of the triangle formed by the base edge and apothem is approximately 7.98 inches.

Now we can find the area of each triangular face:

Area of one triangular face = 1/2 x base x height

Area of one triangular face = 1/2 x 8 x 7.98

Area of one triangular face = 31.92 square inches

Since there are four triangular faces, the total surface area of the pyramid is:

Total surface area = 4 x 31.92

Total surface area = 127.68 square inches

Therefore, the surface area of the pyramid is approximately 127.68 square inches.