Question
3. Find surface area of a square pyramid with "a" of .5cm and "I" of .8cm. (units needed)
Answers
GPT-4o mini
To find the surface area of a square pyramid, we need to calculate the area of the base and the area of the four triangular faces.
1. **Base Area**: The base of the pyramid is a square with side length \( a \).
\[
\text{Base Area} = a^2 = (0.5\, \text{cm})^2 = 0.25\, \text{cm}^2
\]
2. **Triangular Face Area**: The area of one triangular face can be found using the formula for the area of a triangle:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangle is \( a \) (the side of the square base), and the height of the triangle is \( I \) (the slant height).
Therefore,
\[
\text{Area of one triangle} = \frac{1}{2} \times a \times I = \frac{1}{2} \times 0.5\, \text{cm} \times 0.8\, \text{cm} = \frac{1}{2} \times 0.4\, \text{cm}^2 = 0.2\, \text{cm}^2
\]
Since there are four triangular faces:
\[
\text{Total Area of Triangles} = 4 \times 0.2\, \text{cm}^2 = 0.8\, \text{cm}^2
\]
3. **Total Surface Area**: The total surface area of the square pyramid is the sum of the base area and the total area of the triangular faces:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Total Area of Triangles} = 0.25\, \text{cm}^2 + 0.8\, \text{cm}^2 = 1.05\, \text{cm}^2
\]
Therefore, the surface area of the square pyramid is **1.05 cm²**.
1. **Base Area**: The base of the pyramid is a square with side length \( a \).
\[
\text{Base Area} = a^2 = (0.5\, \text{cm})^2 = 0.25\, \text{cm}^2
\]
2. **Triangular Face Area**: The area of one triangular face can be found using the formula for the area of a triangle:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangle is \( a \) (the side of the square base), and the height of the triangle is \( I \) (the slant height).
Therefore,
\[
\text{Area of one triangle} = \frac{1}{2} \times a \times I = \frac{1}{2} \times 0.5\, \text{cm} \times 0.8\, \text{cm} = \frac{1}{2} \times 0.4\, \text{cm}^2 = 0.2\, \text{cm}^2
\]
Since there are four triangular faces:
\[
\text{Total Area of Triangles} = 4 \times 0.2\, \text{cm}^2 = 0.8\, \text{cm}^2
\]
3. **Total Surface Area**: The total surface area of the square pyramid is the sum of the base area and the total area of the triangular faces:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Total Area of Triangles} = 0.25\, \text{cm}^2 + 0.8\, \text{cm}^2 = 1.05\, \text{cm}^2
\]
Therefore, the surface area of the square pyramid is **1.05 cm²**.