How do you find the TSA of a square based pyramid?

The Total Surface Area (TSA) of a square based pyramid is determined by summing the area of its square base and the combined area of its four triangular lateral faces.

Understanding the Components of a Square Based Pyramid

To calculate the TSA, you need to identify two main parts:

• The Base Area: This is the area of the square at the bottom of the pyramid.
• The Lateral Surface Area: This is the sum of the areas of the four triangular faces that meet at the pyramid's apex.

Calculating the Base Area

Since the base is a square, its area is found by squaring the length of one of its sides.

• Let a be the length of the base side.
• Base Area = a²

Calculating the Lateral Surface Area

Each of the four triangular faces is identical. The area of a triangle is calculated as (1/2) * base * height. For the lateral faces of a pyramid:

• The base of each triangle is a (the side length of the square base).
• The height of each triangle is the slant height (l) of the pyramid. The slant height is the distance from the apex of the pyramid down the center of a triangular face to the midpoint of its base.

So, the area of one triangular lateral face is (1/2) * a * l.
Since there are four such faces, the *Lateral Surface Area = 4 (1/2) a l = 2al**.

Formulas for Total Surface Area (TSA)

Combining the base area and the lateral surface area, we get the Total Surface Area (TSA) of a square based pyramid. There are two common formulas, depending on the information you have:

1. When Slant Height (l) is Known

If you know the base side length (a) and the slant height (l), the formula is straightforward:

TSA = Base Area + Lateral Surface Area
TSA = a² + 2al

2. When Vertical Height (h) is Known

Sometimes, the vertical height (h) of the pyramid (the perpendicular distance from the apex to the center of the base) is given instead of the slant height (l). In this case, you can find the slant height using the Pythagorean theorem, which relates h, l, and half of the base side length (a/2).
Learn more about the Pythagorean theorem and geometric principles.

• The relationship is l² = h² + (a/2)².
• Therefore, l = √(h² + (a/2)²) = √(h² + a²/4).

Substitute this expression for l into the first TSA formula:

TSA = a² + 2a√(a²/4 + h²)

Key Variables Defined

Symbol	Definition
TSA	Total Surface Area of the pyramid
a	Length of one side of the square base
l	Slant height of the pyramid (height of a triangular face)
h	Vertical height of the pyramid (from apex to base center)

Example Calculation

Let's find the Total Surface Area (TSA) of a square based pyramid with a base side length (a) of 8 cm and a vertical height (h) of 3 cm.

1. Identify Known Values:

   • a = 8 cm
   • h = 3 cm

2. Choose the Appropriate Formula: Since h is known, we use TSA = a² + 2a√(a²/4 + h²).

3. Calculate the Slant Height (l) first:
   l = √(a²/4 + h²)
l = √(8²/4 + 3²)
l = √(64/4 + 9)
l = √(16 + 9)
l = √25
l = 5 cm

4. Calculate the Base Area:
   Base Area = a² = 8² = 64 cm²

5. Calculate the Lateral Surface Area:
   Lateral Surface Area = 2al = 2 * 8 cm * 5 cm = 80 cm²

6. Calculate the Total Surface Area (TSA):
   TSA = Base Area + Lateral Surface Area
TSA = 64 cm² + 80 cm² = 144 cm²

Thus, the total surface area of the pyramid is 144 cm².