Pyramid Geometry Formulas and Properties

In this article, we can learn about types of pyramids, geometrical formulas for surface area and volume, and their properties.

A pyramid is a three-dimensional solid shape that can have any polygon as its base. Its edges converge at a single vertex or apex.

Its dimensions are defined by the dimensions of the polygon at its base and the length of its lateral edges, which lead to the vertex.

Table of Contents

Types of Pyramids

Triangular pyramid, Square pyramid & Pentagonal pyramid

The type of the pyramid is called according to its base.

The base is triangle type, then it is called Triangular pyramid.

If the base is square, then it is called Square pyramid.

If the base is pentagonal type then it is called Pentagonal pyramid.

Perimeter of a pyramid (p) : It is the distance around the base of the pyramid.

Slant height of a pyramid ( l) : It is a diagonal height from the vertex to center one of the base edges.

Height/Altitude of pyramid ( h) : Altitude ( h) is the perpendicular distance between the apex and bases of a pyramid.

Right Pyramid & Oblique Pyramid

Right Pyramid – The Vertex or apex of the pyramid is directly above the center of its base; then, it is called the right pyramid.

i.e the apex line is perpendicular to the center point of its base.

The square pyramid and triangular pyramid are examples of right pyramids.

Oblique Pyramid – If the Vertex or apex of the pyramid is not directly above the center of its base, then it is called an Oblique pyramid. The face of the oblique pyramid is not congruent.

Basic Pyramid Geometrical formulas

Slant surface or lateral surface area of the pyramid (S)

$S = \frac{1}{2} \times \text{Perimeter of the Base} \times \text{Slant Height}$

Total surface area of the pyramid (SA)

SA = Lateral surface area of the pyramid + Base area of pyramid

Volume of the pyramid (V)

$V = \frac{1}{3} \times \text{Area of the base} \times \text{Height}$

Triangular Pyramid

A triangular pyramid is a polyhedron with a triangular base and the three faces of the triangular pyramid meet at a point known as apex. The triangular pyramid has

A triangular pyramid has one triangular base and three triangular faces, Six edges & Four vertices

The base, faces, edges and vertices are the same for an irregular triangular pyramid or a regular triangular pyramid.

Here,

Height of the triangular pyramid – h

Slant height of the triangular pyramid – s

Base length of the triangular pyramid – a

Surface area and volume of a triangular pyramid

Base Area of a Triangular Pyramid

$A = \frac{1}{2} \times a \times b$

Surface Area of a Triangular Pyramid (SA)

SA = $S A = B a s e A r e a + \frac{1}{2} (P e r i m e t e r \times S l a n t H e i g h t)$

$SA = A + \frac{1}{2} \ (3a + s)$

Volume of a Triangular Pyramid (V)

V = $\frac{1/}{3 x} B a s e A r e a \times H e i g h t$

$V = \frac{1}{3} A h$

$V = \frac{1}{6} abh$

Square Pyramid

A square pyramid is square based three-dimensional geometrical figure. All four sides are triangular and they meet at one single point known as the apex.

A Square pyramid has

a) Five Faces
b) Four Side Faces are Triangles
c) Square Base
d) Five Vertices (corner points)
e) Eight Edges

Base Area of the square pyramid (A)

A = a²

Slant surface or lateral surface area of the pyramid (S)

S= 1/2 x 4a x s = 2as

Surface area of square pyramid (SA)

SA = a² +2as

The volume of a Square Pyramid (V)

$V = \frac{1}{3} a^3 h$

Where ” s ” is the slant height, “a” is the base length, and “h” is the height the of the square pyramid

Pentagonal Pyramid

A pentagonal pyramid is a pyramid with a pentagonal base.

A Pentagonal pyramid has

Five triangular face

Pentagonal base
Six Faces
Six Vertices (corner points)
Ten Edges

Volume of a pentagonal pyramid (V)

$V = \frac{1}{3} \times \text{Area of the pentagonal base} \times \text{height}$

$V = \frac{5}{6} \times a \times b \times h$

Surface Area of a Pentagonal Pyramid (SA)

SA = 5/2 × a(b + s)

$SA = 1.72 \ a^2 + \frac{5}{2} \ a \ h$

Hexagonal Pyramid – A hexagonal pyramid is a pyramid having a hexagonal base. It has a total of 6 sides and 6 triangular lateral faces. A hexagonal pyramid is also known as heptahedron.

Pyramid name	Faces	Edges	Vertices
Triangular Pyramid (or) Tetrahedron	4	6	4
Square Pyramid	5	8	5
Pentagonal Pyramid	6	10	6
Hexagonal Pyramid	7	12	7
Heptagonal Pyramid	8	14	8
Octagonal Pyramid	9	16	9