How do you calculate the total surface area of a pyramid that has a square base with sides measuring 8 units and a slant height of 5 units?

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Total Surface Area of a Pyramid

To find the total surface area of a pyramid with a square base, we need to calculate the area of the base as well as the areas of the triangular faces.

Step 1: Calculate the Area of the Base

The base of our pyramid is a square with side lengths of 8 units. The area Abase of the base can be calculated using the formula:

Abase = side × side = 8 × 8 = 64 	ext{ square units}

Step 2: Calculate the Area of the Triangular Faces

The pyramid has four triangular faces. To find the area of one triangular face, we can use the formula:

Atriangle = 0.5 × base × height

In this case, each triangle has a base of 8 units (the same as the side of the base) and a height equal to the slant height of the pyramid, which is 5 units. Thus, the area of one triangular face is:

Atriangle = 0.5 × 8 × 5 = 20 	ext{ square units}

Since there are four triangular faces, the total area of the triangular faces Atriangles is:

Atriangles = 4 × Atriangle = 4 × 20 = 80 	ext{ square units}

Step 3: Calculate the Total Surface Area

The total surface area Atotal of the pyramid is the sum of the area of the base and the area of the triangular faces:

Atotal = Abase + Atriangles
Atotal = 64 + 80 = 144 	ext{ square units}

Conclusion

Therefore, the total surface area of the pyramid is 144 square units.

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