# Square Pyramid Calculator

## Square Pyramid Shape

h = height

s = slant height

a = side length

e = lateral edge length

r = a/2

V = volume

L = lateral surface area

B = base surface area

A = total surface area

## Calculator Use

This online calculator will calculate the various properties of a square pyramid given 2 known variables. The square pyramid is a special case of a pyramid where the base is square. It is a regular pyramid with a square base.

* **Units:** Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft^{2} or ft^{3}. For example, if you are starting with mm and you know r and h in mm, your calculations will result with s in mm,
V in mm^{3}, L in mm^{2}, B in mm^{2} and A in mm^{2}.

* **NAN:** means not a number. This will show as a result if you are using values that just do not make sense as reasonable values for a pyramid.

Below are the standard formulas for a pyramid. Calculations are based on algebraic manipulation of these standard formulas.

## Square Pyramid Formulas derived in terms of side length a and height h:

**Volume of a square pyramid:****V = (1/3)a**^{2}h

**Slant Height of a square pyramid**:- By the pythagorean theorem we know that
- s
^{2}= r^{2}+ h^{2} - since r = a/2
- s
^{2}= (1/4)a^{2}+ h^{2}, and **s = √(h**^{2}+ (1/4)a^{2})- This is also the height of a triangle side

**Lateral Surface Area of a square pyramid**(4 isosceles triangles):- For the isosceles triangle Area = (1/2)Base x Height. Our base is side length a and for this calculation our height for the triangle is slant height s. With 4 sides we need to multiply by 4.
- L = 4 x (1/2)as = 2as = 2a√(h
^{2}+ (1/4)a^{2}) - Squaring the 2 to get it back inside the radical,
**L = a√(a**^{2}+ 4h^{2})

**Base Surface Area of a square pyramid**(square):**B = a**^{2}

**Total Surface Area of a square pyramid**:- A = L + B = a
^{2}+ a√(a^{2}+ 4h^{2})) **A = a(a + √(a**^{2}+ 4h^{2}))

- A = L + B = a

### Square Pyramid Calculations:

Other formulas for calculations are derived from the formulas above.

## References

Weisstein, Eric W. "Square Pyramid." From* MathWorld*--A Wolfram Web Resource. Square Pyramid.

**Cite this content, page or calculator as:**

Furey, Edward "Square Pyramid Calculator"; from *https://www.calculatorsoup.com* - Online Calculator Resource.