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A regular square pyramid is shown below. What is the lateral area of this pyramid?

To find the lateral area of the regular square pyramid shown in the diagram, we'll follow these steps:

1. Identify the slant height and base perimeter of the pyramid.
2. Use the formula for the lateral area of a pyramid.

The given measurements are:
- The height of the triangular face (not the slant height) is 6 inches.
- The base length of the triangular face is 16 inches.

Since the base is a square with side length \( s \), we need to calculate the perimeter of the base (P):
\[ P = 4s = 4 \times 16 = 64 \text{ inches} \]

To find the slant height (\( l \)), we need to recognize that the height given (6 inches) is the altitude of the triangular face, which forms a right triangle with the slant height and half of the base side length. 

So, half the base length of the pyramid is:
\[ \frac{s}{2} = \frac{16}{2} = 8 \text{ inches} \]

Using the Pythagorean theorem in the right triangle formed by the slant height, altitude, and half the base length:
\[ l = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ inches} \]

Now, use the formula for the lateral area (A) of a pyramid:
\[ A = \frac{1}{2} \times P \times l \]

Plugging in the values, we get:
\[ A = \frac{1}{2} \times 64 \times 10 = \frac{1}{2} \times 640 = 320 \text{ square inches} \]

Thus, the lateral area of the regular square pyramid is:
\[ 320 \text{ square inches} \]

Would you like further details or have any questions? Here are some questions you might want to ask:

1. How do you find the volume of this pyramid?
2. What is the surface area of this pyramid including the base?
3. How do you calculate the height of the pyramid from the slant height?
4. What is the formula for the lateral area of a regular pyramid with a different base shape?
5. How do you determine if a pyramid is regular?

**Tip:** When working with geometric shapes, always verify the type of triangle and use the correct relationships and theorems to find missing dimensions accurately.

Discover how to calculate the lateral area of a regular square pyramid with a step-by-step solution. This problem involves using the perimeter of the base, slant height calculation using the Pythagorean theorem, and applying the formula for the lateral area of pyramids. Suitable for high school geometry students.

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