GATE Civil 2016 Set 2 | GA | Question: 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. $0.25 \: x^{2}$

Comments

Silly mistake :

I calculated Surface area instead of lateral area.

Surface Area: area of all side inclduing base also

Lateral Area: area of all sides excluding base also

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 

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@MiNiPanda and @Thadymademe Apothem or slant height is perpendicular to the side of the base and it bisects the side, on to which it is perpendicular.

Answer 2

 

• Simple way to solve it is [ without having any formula ] 
• We know that pyramid is nothing take a square as base and on top of that put 4 triangles and it is a square 
• Total surface area = Area of square +. 4 triangles area 
• But question is about lateral surafe area, which means we don't include area of square which is base 
• Now what is area of one triangle = 1/2 * base * height  [ to get final answer just multiply by 4 ] 
• It is given that base of perimeter is x so single base is x/4 [ because square 4 sides ] 
• Now height they said x/2 
• So Final answer = 4 * [ 1/2 * x/4 * x/2 ] = x^2 / 4 = 0.25 x^2 

 

Comments

@Venu Gopal you are multipying the slant height to the base , How it is the areaof triagle ?In area of triangle we need that height which is perpendicular to the base , not the slant height .