Answer:
h = height
s = slant height
a = side length
P = perimeter of base
e = lateral edge length
r = a/2
V = volume
L = lateral surface area
B = base surface area
A = total surface area
Step-by-step explanation:
Volume of a square pyramid:
V = (1/3)a2h
Slant Height of a square pyramid:
By the pythagorean theorem we know that
s2 = r2 + h2
since r = a/2
s2 = (1/4)a2 + h2, and
s = √(h2 + (1/4)a2)
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√(h2 + (1/4)a2)
Squaring the 2 to get it back inside the radical,
L = a√(a2 + 4h2)
Base Surface Area of a square pyramid (square):
B = a2
Total Surface Area of a square pyramid:
A = L + B = a2 + a√(a2 + 4h2))
A = a(a + √(a2 + 4h2))