To calculate the surface area of a square pyramid using the formula [tex]\( A = \text{Base Area} + \text{Lateral Surface Area} \)[/tex], we'll break it down into a step-by-step solution.
### Step 1: Determine the Base Area
The base of a square pyramid is a square. The area of a square base is calculated using the formula:
[tex]\[
\text{Base Area} = s^2
\][/tex]
where [tex]\( s \)[/tex] is the length of a side of the square base. For this problem, the side length [tex]\( s \)[/tex] is 5 units.
[tex]\[
\text{Base Area} = 5^2 = 25 \, \text{square units}
\][/tex]
### Step 2: Determine the Lateral Surface Area
The lateral surface area of a square pyramid consists of the areas of the four triangular faces. Each face is an isosceles triangle with a base [tex]\( s \)[/tex] and height [tex]\( l \)[/tex] (the slant height of the pyramid). The area of one such triangle is given by:
[tex]\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Since there are four triangles,
[tex]\[
\text{Lateral Surface Area} = 4 \times \frac{1}{2} \times s \times l
\][/tex]
Simplifying,
[tex]\[
\text{Lateral Surface Area} = 2 \times s \times l
\][/tex]
Given [tex]\( s = 5 \)[/tex] units and [tex]\( l = 10 \)[/tex] units:
[tex]\[
\text{Lateral Surface Area} = 2 \times 5 \times 10 = 100 \, \text{square units}
\][/tex]
### Step 3: Calculate the Total Surface Area
Finally, add the base area and the lateral surface area to get the total surface area of the square pyramid:
[tex]\[
A = \text{Base Area} + \text{Lateral Surface Area}
\][/tex]
[tex]\[
A = 25 \, \text{square units} + 100 \, \text{square units}
\][/tex]
[tex]\[
A = 125 \, \text{square units}
\][/tex]
### Summary
- Base Area: [tex]\( 25 \, \text{square units} \)[/tex]
- Lateral Surface Area: [tex]\( 100 \, \text{square units} \)[/tex]
- Total Surface Area: [tex]\( 125 \, \text{square units} \)[/tex]
Thus, the total surface area of the square pyramid is [tex]\( 125 \, \text{square units} \)[/tex].
