A pyramid has an equilateral triangle as its base, with each side measuring $8 ext{ cm}$. Its slant edge is $24 ext{ cm}$. Find the total surface area of the pyramid in $ ext{cm}^2$.

Question

A pyramid has an equilateral triangle as its base, with each side measuring $8 	ext{ cm}$. Its slant edge is $24 	ext{ cm}$. Find the total surface area of the pyramid in $	ext{cm}^2$.

Accepted Answer

**Solution:**

1. Calculate the area of the equilateral triangle base:
   - Formula: $	ext{Area} = \frac{\sqrt{3}}{4} 	imes (	ext{side})^2$
   - Substitute side = 8 cm:
   $$\frac{\sqrt{3}}{4} 	imes 8^2 = \frac{\sqrt{3}}{4} 	imes 64 = 16\sqrt{3} 	ext{ cm}^2$$

2. Find the height of each triangular face (height of the lateral face):
   - Use Pythagoras theorem with slant edge as hypotenuse:
   $$h^2 = 24^2 - \left(\frac{8}{2}ight)^2 = 576 - 16 = 560$$
   $$h = \sqrt{560} = 4\sqrt{35} 	ext{ cm}$$

3. Calculate the area of one triangular face:
   - Base = 8 cm, height = $4\sqrt{35}$ cm
   $$	ext{Area} = \frac{1}{2} 	imes 8 	imes 4\sqrt{35} = 16\sqrt{35} 	ext{ cm}^2$$

4. Total surface area of the pyramid:
   - Base area + 3 × lateral face area
   $$= 16\sqrt{3} + 3 	imes 16\sqrt{35} = 16\sqrt{3} + 48\sqrt{35} 	ext{ cm}^2$$

**Answer:** $(16\sqrt{3} + 48\sqrt{35}) 	ext{ cm}^2$

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