GATE2016 CE-2: GA-9

Question

A square pyramid has a base perimeter $x$, and the slant height is half of the perimeter. What is the lateral surface area of the pyramid

• A. $x^{2}$
• B. $0.75 x^{2}$
• C. $0.50 x^{2}$
• D. $0.25 x^{2}$

Answer 1

Given, perimeter of a pyramid is $x,$ so slant height is $\dfrac{x}{2}.$

Lateral Surface Area of a pyramid

$= \dfrac{1}{2} \times \text{perimeter}\times \text{slant height}$
$= \dfrac{1}{2} \times x \times \dfrac{x}{2}$
$= 0.25 x^{2}$

so option D is correct .

Comments

please give the source link of this formula, I am not able to find the same formula

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Lateral surface area is total area covered by all 4 triangular faces.

let suppose edge length of our base square$=$$a$

It's given that $4a=x$  and $h=\frac{x}{2}$   // h is slant hight or Apothem

Total area then of 4 triangular faces$=4*(\frac{1}{2}*a*h)$ $=\frac{x^{2}}{4}$

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Is the Apothem or slant height falling perpendicular to the base? I think it is according to the diagram attached. If so then we can see that area of one triangular face is $\frac{1}{2}$*slant_height*one_side_of_square [1/2*height*base].

slant_height=x/2 given

one_side_of_square=x/4 [4a=x => a=x/4]

Since there are 4 triangular faces so, 4*($\frac{1}{2}$*slant_height*one_side_of_square)

= 2*(x/2)*(x/4) = 0.25x²

 

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@MiNiPanda apotherm is side bisector to the base . Not perpendicular to the base