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The great pyramids of Egypt are the most popular real life examples of the shape of a three dimensional pyramid. A pyramid is a 3D shape with a base shaped like a polygon. In this article we will learn the formulas to calculate the surface area and the volume of a triangular pyramid....Read MoreRead Less

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.

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.

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 \(in^3\).

**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.

Frequently Asked Questions

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.