Surface Area of a triangular pyramid | Math, geometry | ShowMe - Free Printable

Problem Analysis:

The image shows a square pyramid with the following dimensions:
- The base is a square with side length \( 10 \, \text{m} \).
- The slant height (the distance from the apex to the midpoint of one of the base edges) is \( 12 \, \text{m} \).
- The area of the base is given as \( 43.3 \, \text{m}^2 \).

The task is to find the volume of the pyramid.

Step-by-Step Solution:

#### Step 1: Verify the area of the base
The base of the pyramid is a square with side length \( 10 \, \text{m} \). The area of a square is given by:
\[
\text{Area of base} = \text{side}^2 = 10^2 = 100 \, \text{m}^2
\]
However, the problem states that the area of the base is \( 43.3 \, \text{m}^2 \). This discrepancy suggests that the base might not be a perfect square or there might be an error in the problem statement. For the sake of solving the problem, we will use the given area of the base, \( 43.3 \, \text{m}^2 \).

#### Step 2: Determine the height of the pyramid
To find the volume of the pyramid, we need the perpendicular height from the apex to the center of the base. Let's denote this height as \( h \).

The slant height (\( 12 \, \text{m} \)) is the hypotenuse of a right triangle formed by:
1. The height of the pyramid (\( h \)).
2. Half the diagonal of the base.

First, calculate the diagonal of the square base. The diagonal \( d \) of a square with side length \( s \) is given by:
\[
d = s\sqrt{2} = 10\sqrt{2} \, \text{m}
\]
Half of the diagonal is:
\[
\frac{d}{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} \, \text{m}
\]

Now, consider the right triangle formed by:
- The height of the pyramid (\( h \)),
- Half the diagonal of the base (\( 5\sqrt{2} \, \text{m} \)),
- The slant height (\( 12 \, \text{m} \)).

Using the Pythagorean theorem:
\[
\text{Slant height}^2 = \text{Height}^2 + \left(\frac{\text{Diagonal}}{2}\right)^2
\]
\[
12^2 = h^2 + (5\sqrt{2})^2
\]
\[
144 = h^2 + 50
\]
\[
h^2 = 144 - 50
\]
\[
h^2 = 94
\]
\[
h = \sqrt{94} \, \text{m}
\]

#### Step 3: Calculate the volume of the pyramid
The volume \( V \) of a pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substitute the given base area (\( 43.3 \, \text{m}^2 \)) and the calculated height (\( \sqrt{94} \, \text{m} \)):
\[
V = \frac{1}{3} \times 43.3 \times \sqrt{94}
\]

First, calculate \( \sqrt{94} \):
\[
\sqrt{94} \approx 9.7 \, \text{m}
\]

Now, substitute this value:
\[
V = \frac{1}{3} \times 43.3 \times 9.7
\]
\[
V = \frac{1}{3} \times 420.01
\]
\[
V \approx 140.0033 \, \text{m}^3
\]

Final Answer:

\[
\boxed{140}
\]