Question
find the surface area of a square pyramid with ''a''of.5cm and;;1;; of.8cm units needed
Answers
Answered by
GPT-4o mini
To find the surface area of a square pyramid, you need to sum the area of the square base and the area of the four triangular faces.
### Given:
- Side length of square base (a): 0.5 cm
- Height of the pyramid (h): 0.8 cm
### Step 1: Area of the base
The area of the square base \( A_{\text{base}} \) is given by:
\[
A_{\text{base}} = a^2 = (0.5 \, \text{cm})^2 = 0.25 \, \text{cm}^2
\]
### Step 2: Area of the triangular faces
To find the area of one triangular face, we need the slant height \( l \). The slant height is the height of the triangle from the center of the base to the apex of the pyramid, and it can be found using the Pythagorean theorem.
1. First, find the length from the center of the base to the midpoint of one side (which is half of \( a \)):
\[
\frac{a}{2} = \frac{0.5 \, \text{cm}}{2} = 0.25 \, \text{cm}
\]
2. Now, use the Pythagorean theorem to find the slant height \( l \):
\[
l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} = \sqrt{(0.8 \, \text{cm})^2 + (0.25 \, \text{cm})^2}
\]
Calculating this:
\[
l = \sqrt{(0.8)^2 + (0.25)^2} = \sqrt{0.64 + 0.0625} = \sqrt{0.7025} \approx 0.8375 \, \text{cm}
\]
3. The area of one triangular face \( A_{\text{triangle}} \) is given by:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times l = \frac{1}{2} \times 0.5 \, \text{cm} \times 0.8375 \, \text{cm} \approx 0.209375 \, \text{cm}^2
\]
4. Since there are four triangular faces, the total area of the triangular faces \( A_{\text{triangles}} \):
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 0.209375 \, \text{cm}^2 \approx 0.8375 \, \text{cm}^2
\]
### Step 3: Total surface area
Now, sum the area of the base and the area of the triangular faces:
\[
\text{Total Surface Area} = A_{\text{base}} + A_{\text{triangles}} = 0.25 \, \text{cm}^2 + 0.8375 \, \text{cm}^2 \approx 1.0875 \, \text{cm}^2
\]
### Conclusion:
The surface area of the square pyramid is approximately **1.0875 cm²**.
### Given:
- Side length of square base (a): 0.5 cm
- Height of the pyramid (h): 0.8 cm
### Step 1: Area of the base
The area of the square base \( A_{\text{base}} \) is given by:
\[
A_{\text{base}} = a^2 = (0.5 \, \text{cm})^2 = 0.25 \, \text{cm}^2
\]
### Step 2: Area of the triangular faces
To find the area of one triangular face, we need the slant height \( l \). The slant height is the height of the triangle from the center of the base to the apex of the pyramid, and it can be found using the Pythagorean theorem.
1. First, find the length from the center of the base to the midpoint of one side (which is half of \( a \)):
\[
\frac{a}{2} = \frac{0.5 \, \text{cm}}{2} = 0.25 \, \text{cm}
\]
2. Now, use the Pythagorean theorem to find the slant height \( l \):
\[
l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} = \sqrt{(0.8 \, \text{cm})^2 + (0.25 \, \text{cm})^2}
\]
Calculating this:
\[
l = \sqrt{(0.8)^2 + (0.25)^2} = \sqrt{0.64 + 0.0625} = \sqrt{0.7025} \approx 0.8375 \, \text{cm}
\]
3. The area of one triangular face \( A_{\text{triangle}} \) is given by:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times l = \frac{1}{2} \times 0.5 \, \text{cm} \times 0.8375 \, \text{cm} \approx 0.209375 \, \text{cm}^2
\]
4. Since there are four triangular faces, the total area of the triangular faces \( A_{\text{triangles}} \):
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 0.209375 \, \text{cm}^2 \approx 0.8375 \, \text{cm}^2
\]
### Step 3: Total surface area
Now, sum the area of the base and the area of the triangular faces:
\[
\text{Total Surface Area} = A_{\text{base}} + A_{\text{triangles}} = 0.25 \, \text{cm}^2 + 0.8375 \, \text{cm}^2 \approx 1.0875 \, \text{cm}^2
\]
### Conclusion:
The surface area of the square pyramid is approximately **1.0875 cm²**.