Question
A monument in a park is shaped like a square pyramid. The dimensions are shown in the net.
What is the surface area of the monument, including the base?
Responses
4 m²
4 m²
12 m²
12 m²
14 m²
14 m²
24 m²
24 m²
An unfolded square base pyramid with the sides of the base being two meters and an arrow points from the label two point five meters to a dotted line representing the length of the pyramid.
What is the surface area of the monument, including the base?
Responses
4 m²
4 m²
12 m²
12 m²
14 m²
14 m²
24 m²
24 m²
An unfolded square base pyramid with the sides of the base being two meters and an arrow points from the label two point five meters to a dotted line representing the length of the pyramid.
Answers
GPT-4o mini
To find the surface area of a square pyramid including the base, we need to calculate the area of the base and the area of the four triangular sides.
1. **Area of the Base**:
The base is a square, and if each side of the base is 2 meters, the area of the base is:
\[
\text{Area}_{\text{base}} = \text{side}^2 = 2\, \text{m} \times 2\, \text{m} = 4\, \text{m}^2
\]
2. **Area of the Triangular Sides**:
The pyramid has four triangular sides. The area of one triangular side can be calculated using the formula:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangular side is the same as the side of the square base, which is 2 meters. The height of the triangular side is the slant height, which is given as 2.5 meters.
Therefore, the area of one triangular side is:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 2\, \text{m} \times 2.5\, \text{m} = \frac{1}{2} \times 5\, \text{m}^2 = 2.5\, \text{m}^2
\]
Since there are four triangular sides, the total area of the triangles is:
\[
\text{Total Area}_{\text{triangles}} = 4 \times 2.5\, \text{m}^2 = 10\, \text{m}^2
\]
3. **Total Surface Area**:
Now, we can add the area of the base and the total area of the triangular sides:
\[
\text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{triangles}} = 4\, \text{m}^2 + 10\, \text{m}^2 = 14\, \text{m}^2
\]
Thus, the surface area of the monument, including the base, is **14 m²**.
1. **Area of the Base**:
The base is a square, and if each side of the base is 2 meters, the area of the base is:
\[
\text{Area}_{\text{base}} = \text{side}^2 = 2\, \text{m} \times 2\, \text{m} = 4\, \text{m}^2
\]
2. **Area of the Triangular Sides**:
The pyramid has four triangular sides. The area of one triangular side can be calculated using the formula:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangular side is the same as the side of the square base, which is 2 meters. The height of the triangular side is the slant height, which is given as 2.5 meters.
Therefore, the area of one triangular side is:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 2\, \text{m} \times 2.5\, \text{m} = \frac{1}{2} \times 5\, \text{m}^2 = 2.5\, \text{m}^2
\]
Since there are four triangular sides, the total area of the triangles is:
\[
\text{Total Area}_{\text{triangles}} = 4 \times 2.5\, \text{m}^2 = 10\, \text{m}^2
\]
3. **Total Surface Area**:
Now, we can add the area of the base and the total area of the triangular sides:
\[
\text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{triangles}} = 4\, \text{m}^2 + 10\, \text{m}^2 = 14\, \text{m}^2
\]
Thus, the surface area of the monument, including the base, is **14 m²**.