To find the various properties of a square pyramid with a base edge of 12 units and a height (H) of 18 units, we can calculate the following:
1. Volume (V)
The volume \( V \) of a square pyramid can be calculated using the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
First, we calculate the base area \( A \):
\[ A = \text{Base Edge}^2 = 12^2 = 144 \text{ square units} \]
Now we can find the volume:
\[ V = \frac{1}{3} \times 144 \times 18 = \frac{2592}{3} = 864 \text{ cubic units} \]
2. Surface Area (SA)
The surface area \( S \) of a square pyramid is given by the formula:
\[ S = \text{Base Area} + \text{Lateral Surface Area} \]
The lateral surface area consists of four triangular faces. The area of one triangular face is:
\[ A_{\text{triangle}} = \frac{1}{2} \times \text{Base Edge} \times \text{Slant Height} \]
To find the slant height \( l \), we can use the Pythagorean theorem. The slant height forms a right triangle with half the base edge and the height of the pyramid:
\[ l = \sqrt{\left(\frac{\text{Base Edge}}{2}\right)^2 + \text{Height}^2} \]
Calculating the half base edge:
\[ \frac{12}{2} = 6 \]
Now, using the Pythagorean theorem:
\[ l = \sqrt{6^2 + 18^2} = \sqrt{36 + 324} = \sqrt{360} = 6\sqrt{10} \text{ units} \]
Now we can find the area of one triangular face:
\[ A_{\text{triangle}} = \frac{1}{2} \times 12 \times 6\sqrt{10} = 36\sqrt{10} \text{ square units} \]
The total lateral surface area (4 triangles) is:
\[ \text{Lateral Surface Area} = 4 \times 36\sqrt{10} = 144\sqrt{10} \text{ square units} \]
Now, we can calculate the total surface area:
\[ S = \text{Base Area} + \text{Lateral Surface Area} = 144 + 144\sqrt{10} \]
Thus, the surface area is:
\[ S = 144 + 144\sqrt{10} \text{ square units} \]
Summary
- Volume: \( 864 \) cubic units
- Surface Area: \( 144 + 144\sqrt{10} \) square units
If you need any further calculations or explanations, feel free to ask!