Question

Determine the surface area of a pyramid with a triangular base. The triangle base has two equal sides of 13 in., and the height of the base side is 12 in. The slant height of the pyramid is 7 in.(1 point)
in2

Answer 1

To find the surface area of a pyramid with a triangular base, we need to calculate the area of the triangular base and the area of the three triangular faces.

### Step 1: Calculate the area of the triangular base

The base of the pyramid is a triangle with two equal sides (an isosceles triangle) and the height from the apex of the triangle to the base side. The formula for the area \( A \) of a triangle is:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

For our isosceles triangle, we need to find the base. In this case, the height divides the base into two equal segments. Let the base be \( b \).

Using the Pythagorean theorem on one of the right triangles formed by the height (12 in) and half the base:

\[
(13)^2 = (b/2)^2 + (12)^2
\]

Calculating this:

\[
169 = (b/2)^2 + 144
\]
\[
(b/2)^2 = 169 - 144
\]
\[
(b/2)^2 = 25
\]
\[
b/2 = 5 \quad \Rightarrow \quad b = 10
\]

Now we can calculate the area of the triangular base:

\[
A = \frac{1}{2} \times 10 \times 12 = \frac{120}{2} = 60 \text{ in}^2
\]

### Step 2: Calculate the area of the triangular faces

The pyramid has three triangular faces. The area of each triangular face can be calculated using the formula:

\[
A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{height}
\]

**For the two triangular faces with the equal sides:**

Each of these faces has a base equal to one side of the triangular base (10 in) and a slant height of 7 in.

Calculating the area of one of those faces:

\[
A_\text{face1} = \frac{1}{2} \times 10 \times 7 = \frac{70}{2} = 35 \text{ in}^2
\]

Since there are two such triangular faces:

\[
\text{Total area of two faces} = 2 \times 35 = 70 \text{ in}^2
\]

**For the triangular face opposite the base:**

This face has a base equal to one side of the triangular base (10 in) and a slant height of 7 in.

Since we already calculated this section, the area of the last triangular face would also be:

\[
A_\text{face2} = \frac{1}{2} \times 10 \times 7 = 35 \text{ in}^2
\]

### Step 3: Calculate total surface area

Now we find the total surface area by adding the area of the base and the areas of the three triangular faces:

\[
\text{Total Surface Area} = A_\text{base} + A_\text{face1} + A_\text{face2}
\]

With two faces having the same dimensions, we compute:

\[
\text{Total Surface Area} = 60 \ + \ (35 + 35 + 35) = 60 + 105 = 165 \text{ in}^2
\]

Thus, the total surface area of the pyramid is:

\[
\boxed{165} \text{ in}^2
\]