Formulas Related to the Triangular Pyramid
A triangular pyramid is a solid shape with a triangular base and three triangular lateral faces. The lateral faces meet at a common point known as the apex of the pyramid.
A triangular pyramid has two main formulas – one used to calculate the surface area and the other used to calculate its volume.
A triangular pyramid as a solid shape is as shown in the image.
Formula for the Surface Area of a Triangular Pyramid
The surface area of a pyramid is the sum of the areas of all its lateral faces and the base. A triangular pyramid has 3 lateral faces and 1 base, all triangular in shape. So its total surface area is given by:
Total surface area = Lateral surface area + Base surface area
Here the lateral surface area is the sum of the areas of the three triangular lateral faces.
Surface area is always measured in square units.
Formula for the Volume of a Triangular Pyramid
Volume of a triangular pyramid is the measure of the space enclosed within the faces of the pyramid.
Volume of a triangular pyramid, V = \(\left(\frac{1}{3}\right)\) × Base Area × h
In which,
- Base area is the area of the triangular base
- ‘h’ is the height of the pyramid. It is the perpendicular distance from the apex to the base.
Volume is measured in cubic units such as \(m^3,cm^3\),and \(in^3\).
Solved Examples
Example 1: What is the volume of a triangular pyramid which has a base area of 4 \(cm^2\) and the height is 5 centimeters?
Solution:
Details in the question,
Base area = 4 \(cm^2\)
Height = 5 centimeters
V= \(\frac{1}{3}\) × Base Area × Height Formula for the volume of a triangular pyramid
Substituting the values:
V = \(\frac{1}{3}\) × 4 × 5
V = 6.67 Simplify
Hence, the volume of the given triangular pyramid is 6.67 \(cm^3\).
Example 2: Joe made a triangular pyramid with sand near the ocean. Its volume is 610 \(cm^3\) and the base area is 61 square centimeters. What is the height of the triangular pyramid?
Solution:
Given data, base area = 61 \(cm^2\)
Volume = 610 \(cm^3\)
V= \(\frac{1}{3}\) × Base Area × Height Formula for the volume of a triangular pyramid
Substituting the given values:
We get, \(\frac{1}{3}\) × 61 × h = 610
Solving for h,
h = \(\frac{610\times 3}{61}\)
h = 30 [Simplify]
Hence, the height of the triangular pyramid is 30 cm.
Example 3:
Determine the surface area of the triangular pyramid given in the diagram.
Solution:
To determine the surface area of the given pyramid let us draw its net:
Area of the base: \(\frac{1}{2}\) × 9 × 6 = 27
Area of the lateral face: \(\frac{1}{2}\) × 9 × 11 = 49.5
The surface area of the triangular pyramid = Areas of the lateral faces + Area of the base
= 27 + 49.5 + 49.5 + 49.5 [There are three identical lateral faces]
= 175.5
Therefore, the surface area of the triangular pyramid is 175.5 square inches.
A pyramid having its apex aligned with the center of its base is known as a right pyramid.
A right pyramid with a regular polygon shaped base is known as a regular pyramid.
Like any other solid shape the volume of a triangular pyramid is measured in cubic units.
A polyhedron with four triangular faces, six straight line edges and four vertices is known as a tetrahedron. So, a tetrahedron is just another term for a triangular pyramid.