Understanding the properties of a square pyramid is crucial for calculating its lateral surface area. A square pyramid is a three-dimensional geometric shape with a square base and four triangular faces that meet at a single point (apex).

The problem provides us with two key pieces of information about the pyramid:

• The base perimeter is given as $x$.
• The slant height is stated as half of the perimeter, which means $\frac{x}{2}$.

Pyramid Base Side Calculation

First, we need to determine the side length of the square base. For a square, all four sides are equal in length. The perimeter of a square is the sum of its four sides.

Let $a$ be the side length of the square base.

Given: Perimeter $= x$

Formula for perimeter of a square: $\text{Perimeter} = 4 \times \text{side}$

So, $x = 4a$

To find the side length, divide the perimeter by 4:

$a = \frac{x}{4}$

Lateral Surface Area Formula

The lateral surface area of a pyramid is the sum of the areas of all its triangular faces. For a square pyramid, there are four identical triangular faces. The area of one such triangular face is calculated using the formula for the area of a triangle: $\frac{1}{2} \times \text{base} \times \text{height}$.

In this context:

• The base of each triangular face is the side length of the square base, $a$.
• The height of each triangular face is the slant height, $l$.

The area of one triangular face $= \frac{1}{2} \times a \times l$.

Since there are four such faces, the total lateral surface area (LSA) of the square pyramid is:

$\text{LSA} = 4 \times \left(\frac{1}{2} \times a \times l\right)$

$\text{LSA} = 2 \times a \times l$

Surface Area Calculation for Pyramid

Now, we substitute the values we have found for the side length $a$ and the given slant height $l$ into the lateral surface area formula.

• Side length $a = \frac{x}{4}$
• Slant height $l = \frac{x}{2}$

Substitute these values into the formula for lateral surface area:

$\text{LSA} = 2 \times \left(\frac{x}{4}\right) \times \left(\frac{x}{2}\right)$

Multiply the terms:

$\text{LSA} = 2 \times \frac{x \times x}{4 \times 2}$

$\text{LSA} = 2 \times \frac{x^2}{8}$

Simplify the expression by dividing the numerator and denominator by 2:

$\text{LSA} = \frac{x^2}{4}$

To express this as a decimal, we convert the fraction:

$\text{LSA} = 0.25x^2$

Summary of Lateral Surface Area

The lateral surface area of the square pyramid, given its base perimeter $x$ and slant height as half the perimeter, is $0.25x^2$.

This result matches option 4.