Question

An illustration shows a 3 D triangular prism. The side of the base is 4.5 centimeters. The perpendicular height of the triangular base is 3.9 centimeters. The perpendicular height of the lateral face from the apex to the center of the base is 3.5 centimeters. There are two right angle symbols to show the perpendicular lines. The heights are shown as dashed lines.
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)

Answer 1

To find the surface area of a triangular pyramid (or tetrahedron), we can use the formula for the surface area, which is given by:

\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]

1. **Calculate the Base Area:**
The base is a triangle. The area of a triangle can be calculated using the formula:

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

Given that the base of the triangle is \(4.5\) cm and the height is \(3.9\) cm, we can substitute these values into the formula:

\[
\text{Base Area} = \frac{1}{2} \times 4.5 \times 3.9 = \frac{1}{2} \times 17.55 = 8.775 \text{ cm}^2
\]

2. **Calculate the Lateral Area:**
The lateral area consists of three triangular faces that extend from each edge of the base to the apex of the pyramid. Each of these triangular faces has a base that is equal to each side of the triangular base, and the height is the perpendicular height from the apex to the center of the base. Here we will calculate the area of each triangular face.

Let's denote the three sides of the triangular base as \(a\), \(b\), and \(c\). For our base triangle, if it's an equilateral triangle, all sides are \(4.5\) cm. If they are different, we take the specific values for calculation. Assuming they are equal (as an equilateral triangle) for simplicity:

**Lateral Area of one triangular face:**

Each triangular face has a base of \(4.5\) cm and a height to the apex of \(3.5\) cm, thus:

\[
\text{Area of one lateral face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4.5 \times 3.5 = \frac{1}{2} \times 15.75 = 7.875 \text{ cm}^2
\]

Since there are three such triangular faces, the total lateral area is:

\[
\text{Lateral Area} = 3 \times 7.875 = 23.625 \text{ cm}^2
\]

3. **Calculate the Total Surface Area:**

Now we sum the base area and the lateral area to find the total surface area:

\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 8.775 + 23.625 = 32.4 \text{ cm}^2
\]

Thus, the surface area of each triangular pyramid is **32.4 cm²**.