Pyramid Geometry Formulas and Properties

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

A pyramid is a three dimensional solid shape which can have any polygon as its base and its edges converge a single vertex or apex.

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

Types of Pyramid

Triangular pyramid, Square pyramid & Pentagonal pyramid

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

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

If the base is square type 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 a 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 right pyramid.

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

The square pyramid, triangular pyramid are examples of a right pyramid.

Oblique Pyramid – The Vertex or apex of the pyramid is not having directly above the center of its base, then it is called 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)

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

Total surface area of the pyramid (SA)

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

Volume of the pyramid (V)

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

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

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 the four sides are triangular and they are meet at one single point known as 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 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

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 of the of the square pyramid

Pentagonal Pyramid

A pentagonal pyramid is a pyramid having 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)

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

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

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