SURFACE AREA OF SQUARE PYRAMID FORMULA

Question

SA=LW+L√((W/2)²+H²)+W√((L/2)²+H²)

Please help, I need equations that will solve for the following: LENGTH, L, WIDTH, W, HEIGHT, H of the square pryamid's surface area.

 

Urgent!

Answer 1

Not for the weak at heart....

 

Solving for L....

 

Let k = sqrt( (w/2)^2 + h^2 )

 

The formula becomes:

   S = L( w + sqrt(k)) + (w/2) sqrt ( L^2 + 4H^2)  after factoring out L from the first two terms and

                                                                            factoring out 1/4 from under the radical of the last term

 

 

 ISOLATING the radical :

 

2 [ S - L ( w + sqrt(k)]/w = sqrt( L^2 + 4H^2)

 

 

Squaring both sides:

 

[ 2 [ S - L ( w + sqrt(k)]/w ]^2 = L^2 + 4H^2

 

 

Squaring with FOIL and moving everything to left side,

you get a quadratic equation in terms of L:

 

with A =  { 4 (w + sqrt(k))^2 - w}

       B = 16 s ( w + sqrt(k) )

 

      C = 4s^2 - 4h^2w^2

 

 

Per the quadratic formula, the formula for L:

 

   [8st +or-  sqrt (  64 s t^2 - 16 s (t^2 - w)(4s^2 - 4h^2 w^2) )]/ [8(t^2 - w)]

 

where t = w+sqrt(k)

 

which simplifies to :

 

[2st +or- sqrt (4 s t^2 -  s (t^2 - w)(4s^2 - 4h^2 w^2) )]/ [2 (t^2 - w)]  

 

 

You will get a similar behavior during the solution process and similar results for W because

the formula is same structure for W in terms of L and H.

 

In fact you might be able to reverse the L and W in this formula to save a lot of time, energy,and work.

You will start be replace W with L for k and T. The rest of the steps should be the same.