Question

All edges of a square pyramid are of equal lengths. If the length of the base edge is $20cm$, what is the total surface area?

• A. $400\sqrt{3}$
• B. $400(1+\sqrt{3})$
• C. $100(1+\sqrt{3})$
• D. $400(3+\sqrt{3})$

Answer 1

The correct option is B

$400(1+\sqrt{3})$

Total surface area of the pyramid = The lateral surface area of a square pyramid + Area of the square face of the pyramid

$\Rightarrow$ T.S.A of pyramid = $4\times$ Area of a triangular face of the pyramid + Area of the square face of the pyramid

$\Rightarrow$ T.S.A of pyramid = $4\times \frac{1}{2}\times Baseedgeofthetriangle\times Slantheightofthepyramid$ + Area of the square face of the pyramid ......(i)

Given that Length of the base edge= Length of the lateral edge= $20cm$.
Thus, Base area = $20\times 20=400$

Note that Slant height of the pyramid= $\sqrt{(Lateraledge{)}^2-(Halfthebaseedge{)}^2}=\sqrt{(20{)}^2-(10{)}^2}=10\sqrt{3}$

Now, Lateral surface area of the pyramid = $2\times 20\times 10\sqrt{3}=400\sqrt{3}$

Substituting the values in (i)

Total surface area of the pyramid = $400+400\sqrt{3}=400(1+\sqrt{3})$