What is the total surface area of a pyramid whose base is a square with side $8 ext{ cm}$ and height of the pyramid is $3 ext{ cm}$?

Question

What is the total surface area of a pyramid whose base is a square with side $8 	ext{ cm}$ and height of the pyramid is $3 	ext{ cm}$?

Accepted Answer

**Solution:**

1. Calculate the slant height $l$ using the Pythagorean theorem:
   
   $$l = \sqrt{h^2 + \left(\frac{s}{2}ight)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 	ext{ cm}$$

2. Calculate the area of the square base:
   
   $$A_{base} = s^2 = 8^2 = 64 	ext{ cm}^2$$

3. Calculate the lateral surface area (sum of four triangular faces):
   
   $$A_{lateral} = 4 	imes \left(\frac{1}{2} 	imes s 	imes light) = 2 	imes s 	imes l = 2 	imes 8 	imes 5 = 80 	ext{ cm}^2$$

4. Calculate the total surface area:
   
   $$A_{total} = A_{base} + A_{lateral} = 64 + 80 = 144 	ext{ cm}^2$$

**Answer:** $144 	ext{ cm}^2$

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