Question

Find the surface area of the square pyramid
Height: 4 cm
Base edge: 6 cm

Answer 1

To calculate the surface area of a square pyramid, we need to find both the area of the base and the area of the triangular sides (lateral surface area).

1. **Base Area**:
The base of the pyramid is a square.
\[
\text{Base Area} = \text{base edge} \times \text{base edge} = 6 \, \text{cm} \times 6 \, \text{cm} = 36 \, \text{cm}^2
\]

2. **Lateral Surface Area**:
A square pyramid has 4 triangular faces. First, we need to find the slant height of the pyramid (the height of each triangular face). The slant height can be found using the Pythagorean theorem.

- Half of the base edge:
\[
\frac{6 \, \text{cm}}{2} = 3 \, \text{cm}
\]

- The height of the pyramid is 4 cm. We can use these dimensions to calculate the slant height (\(l\)):
\[
l = \sqrt{\left(\frac{\text{base edge}}{2}\right)^2 + \text{height}^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
\]

Now we can calculate the area of one triangular face:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \, \text{cm} \times 5 \, \text{cm} = \frac{30}{2} = 15 \, \text{cm}^2
\]

Since there are 4 triangular faces:
\[
\text{Total Lateral Surface Area} = 4 \times 15 \, \text{cm}^2 = 60 \, \text{cm}^2
\]

3. **Total Surface Area**:
Now, combine the base area and the lateral surface area to find the total surface area:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 36 \, \text{cm}^2 + 60 \, \text{cm}^2 = 96 \, \text{cm}^2
\]

So, the total surface area of the square pyramid is \( \boxed{96 \, \text{cm}^2} \).