Question : A right square pyramid having a lateral surface area is 624 cm2. If the length of the diagonal of the square is $24 \sqrt{2}$, then the volume of the pyramid is:
Option 1: 1150 cm3
Option 2: 780 cm3
Option 3: 1083 cm3
Option 4: 960 cm3
Correct Answer: 960 cm 3
Solution :
Given: Length of the diagonal of the square = $24\sqrt2$
Let the side of the square be $a$.
We know that,
Length of the diagonal of the square = $a\sqrt2$
Thus, $24\sqrt2=a\sqrt2$
$\therefore a = 24$ cm
Perimeter of square = 24 × 4 = 96 cm
Area of the square = 24 × 24 = 576 cm
2
Given: Lateral surface area of pyramid = 624 cm
2
We know that,
Lateral surface area of pyramid = $\frac{1}{2}$ × perimeter of base × slant height
⇒ $624 = \frac{1}{2}×96$ × slant height
$\therefore$ Slant height = $\frac{624\times2}{96}= 13$ cm
Now, Height = $\sqrt{\text{slant height}^2-{(\frac{\text{side of square}}{2}})^2}$
= $\sqrt{13^2-12^2}$
= $\sqrt{169-144}$
= $5$ cm
Volume of the pyramid = $\frac{1}{3}$× area of base × height
= $\frac{1}{3}\times576\times5$
= $960$ cm
3
Hence, the correct answer is 960 cm
3
.
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