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) cm2
Answers
GPT-4o mini
To find the surface area of each triangular pyramid (also known as a tetrahedron with a triangular base), we first need to determine the surface area formula for the pyramid.
The surface area \( S \) of a triangular pyramid is given by the formula:
\[
S = \text{Area of base} + \text{Area of lateral faces}
\]
1. **Calculate the Area of the Base:**
The base is a triangle, and the area \( A \) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
From the information given:
- Side of the base = 4.5 cm (we can assume this is the base of the triangle).
- Perpendicular height of the triangular base = 3.9 cm.
Therefore,
\[
A = \frac{1}{2} \times 4.5 \times 3.9 = 8.775 \text{ cm}^2
\]
2. **Calculate the Area of the Lateral Faces:**
The lateral faces are triangles forming the sides of the pyramid. Each face can be considered as a triangle with:
- Base = side of the triangular base = 4.5 cm.
- The height of each lateral face can be derived from the apex to the midpoint of the base's side and we can consider the triangular side to have a height of 3.5 cm.
The area of one lateral face can also be calculated as:
\[
A_{\text{lateral}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here,
- Base = 4.5 cm,
- Height = 3.5 cm.
Therefore,
\[
A_{\text{lateral}} = \frac{1}{2} \times 4.5 \times 3.5 = 7.875 \text{ cm}^2
\]
Since there are three lateral faces, the total area of the lateral faces is:
\[
A_{\text{total lateral}} = 3 \times 7.875 = 23.625 \text{ cm}^2
\]
3. **Combine the Areas:**
Now, we can find the total surface area of the pyramid:
\[
S = \text{Area of base} + \text{Area of lateral faces}
\]
\[
S = 8.775 + 23.625 = 32.4 \text{ cm}^2
\]
Therefore, the surface area of each pyramid is **32.4 cm²**.
The surface area \( S \) of a triangular pyramid is given by the formula:
\[
S = \text{Area of base} + \text{Area of lateral faces}
\]
1. **Calculate the Area of the Base:**
The base is a triangle, and the area \( A \) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
From the information given:
- Side of the base = 4.5 cm (we can assume this is the base of the triangle).
- Perpendicular height of the triangular base = 3.9 cm.
Therefore,
\[
A = \frac{1}{2} \times 4.5 \times 3.9 = 8.775 \text{ cm}^2
\]
2. **Calculate the Area of the Lateral Faces:**
The lateral faces are triangles forming the sides of the pyramid. Each face can be considered as a triangle with:
- Base = side of the triangular base = 4.5 cm.
- The height of each lateral face can be derived from the apex to the midpoint of the base's side and we can consider the triangular side to have a height of 3.5 cm.
The area of one lateral face can also be calculated as:
\[
A_{\text{lateral}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here,
- Base = 4.5 cm,
- Height = 3.5 cm.
Therefore,
\[
A_{\text{lateral}} = \frac{1}{2} \times 4.5 \times 3.5 = 7.875 \text{ cm}^2
\]
Since there are three lateral faces, the total area of the lateral faces is:
\[
A_{\text{total lateral}} = 3 \times 7.875 = 23.625 \text{ cm}^2
\]
3. **Combine the Areas:**
Now, we can find the total surface area of the pyramid:
\[
S = \text{Area of base} + \text{Area of lateral faces}
\]
\[
S = 8.775 + 23.625 = 32.4 \text{ cm}^2
\]
Therefore, the surface area of each pyramid is **32.4 cm²**.