Surface Area Of Square Pyramid

Square Pyramid Calculator

Square Pyramid Shape

Pyramid Diagram with h = height, l = length and w = width and s = slant height

h = height
south = slant peak
a = side length
P = perimeter of base
e = lateral edge length
r = a/2
V = book
Fifty = lateral surface area
B = base surface area
A = full surface area
m = h/r = ascension/run = side face gradient
θ = tan^-1(h/r) × 180/π = side confront angle

Calculator Use

This online computer volition 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 since information technology has a foursquare base which is a regular polygon. This is also a right foursquare pyramid where "right" refers to the fact that the noon lies directly above the centroid of the base. In other words the point at the top of the pyramid is directly above the heart bespeak of the foursquare base.

Units: Note that units are shown for convenience merely do not impact the calculations. The units are in identify to give an indication of the social club of the results such as ft, ft² or ft³. For example, if y'all are starting with mm and you know r and h in mm, your calculations will result with s in mm, V in mmⁱⁱⁱ, L in mm^two, B in mm² and A in mm².

NAN: means non a number. This will prove as a effect if you are using values that just practise non make sense equally 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 top = h:

Volume of a Foursquare Pyramid

V = (1/iii)a²h

Slant Height of a square pyramid

By the pythagorean theorem we know that
s² = r^two + h²
since r = a/ii
s² = (i/4)a^two + h², and
s = √(h² + (one/4)a^two)
This is likewise the height of a triangle side

Lateral Surface Area of a square pyramid (× 4 isosceles triangles)

For the isosceles triangle Area = (one/2)Base of operations x Top. Our base is side length a and for this calculation our height for the triangle is slant pinnacle due south. With iv sides we demand to multiply by 4.
L = 4 x (ane/2)as = 2as = 2a√(h² + (1/4)a²)
Squaring the two to get it back within the radical,
Fifty = a√(aⁱⁱ + 4hⁱⁱ)

Base Surface Expanse of a square pyramid (square)

B = a²

Total Surface Area of a square pyramid

A = 50 + B = a² + a√(a² + 4h²))
A = a(a + √(aⁱⁱ + 4h²))

Slope of Pyramid Side Face

To find the pyramid slope of the side face up we want to summate the slope of the line s = slant height
We know that the slope of a line is m = rise/run
For the line southward the rising is h = top of the pyramid
r = a/2 and this is the run equally it forms a correct bending where r meets h at the heart of the base
m = h/(a/2) - in terms of h and a
m = h/r - in terms of h and r

Angle of Pyramid Side Face

The angle of the pyramid side face is the angle formed between the side face and the base
Let's name theta θ = Side Face Angle and alpha α = the correct bending (90°) formed by h and r
Using the Law of Sines we can say that s/sin(α) = h/sin(θ)
Solving for the unknown θ we take
θ = sin^-1[ (h × sin(α)) / s ]
We have another formula for θ in terms of the tangent from trigonometric ratios
Since tan(θ) = side opposite θ / side adjacent θ we can say
tan(θ) = h/r
Solving for the unknown θ
θ = tan^-1(h/r)
θ in both calculations is in radians. Convert radians to degrees by multiplying θ by 180/π