Triangular Pyramid Formula

Volume of a triangular pyramid is found using the formula V = 1/3A.H. A triangular pyramid, also known as a tetrahedron, is a type of pyramid with a triangular base and three triangular faces that meet at a single point called the apex.

In this article, we will learn about, Pyramid Definition, Triangular Pyramid Definition, Triangular Pyramid Formula, Examples and others in detail.

A pyramid is classified into various kinds based on the shape of the base, such as a triangular pyramid, a square pyramid, a pentagonal pyramid, a hexagonal pyramid, etc. An apex is a meeting point of the lateral surfaces or the side faces of a pyramid. The perpendicular distance from the apex of a pyramid to the centre of its base is the height or altitude of a pyramid. The perpendicular distance between the apex and the base of a lateral surface slant height of a pyramid.

Triangular Pyramid is a pyramid that has a triangle as its base. It is also known as a tetrahedron and has three triangular-shaped faces and one triangular base, where the triangular base can be scalar, isosceles, or an equilateral triangle. A triangular is further classified into three types i.e., a regular triangular pyramid, an irregular triangular pyramid, and a right triangular pyramid.

• Regular Triangular Pyramid: A triangular pyramid whose four faces are equilateral triangles is called a regular triangular pyramid. As the pyramid is made up of equilateral triangles, the measure of all its internal angles is 60°.

• Irregular Triangular Pyramid: An irregular triangular pyramid is one whose edges of the base are not equal, i.e., the base of an irregular triangular pyramid is either a scalene triangle or an isosceles triangle. All triangular pyramids are assumed to be regular triangular pyramids unless a triangular pyramid is specifically mentioned as irregular.

• Right Triangular Pyramid: A right triangular pyramid is one whose base is a right-angled triangle and whose apex is aligned above the center of the base.

There are two formulas for a triangular pyramid: the surface area of a triangular pyramid and the volume of a triangular pyramid.

• Surface Area of a Triangular Pyramid
  • Lateral Surface Area of a Triangular Pyramid
  • Total Surface Area of a Triangular Pyramid
• Volume of a Triangular Pyramid

Surface area of a pyramid has two types of surface areas, namely: the lateral surface area and the total surface area, where the surface area of a pyramid is the sum of the areas of the lateral surfaces, or side faces, and the base area of a pyramid.

Lateral Surface Area of a Triangular Pyramid

Lateral Surface Area of a Triangular Pyramid is calculated using the formula:

Lateral Surface Area of a Triangular Pyramid (LSA) = ½ × Perimeter × Slant Height

Total Surface Area of a Triangular Pyramid

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

So, TSA = ½ × perimeter × slant height + ½ × base × height

Total Surface Area of a Triangular Pyramid (TSA) = ½ ×  P ×  l + ½ bh

where,

• P is Perimeter of Base
• l is Slant Height of Pyramid
• b is Base of Triangle at Base
• h is Height of Pyramid

Volume of a pyramid is the total space enclosed between all the faces of a pyramid. The volume of a pyramid is generally represented by the letter “V”, and its formula is equal to one-third of the product of the base area and the height of the pyramid. 

The formula for the volume of a pyramid is given as follows,

Volume of a Triangular Pyramid = 1/3 × base area × height

V = 1/3 × AH cubic units

where,

• V is Volume of Pyramid
• A is Area of Base of a Pyramid
• H is Height or Altitude of a Pyramid

The formula for the volume of a regular triangular pyramid is given as follows

Volume of Regular Triangular Pyramid = a³/6√2 cubic units

Where a isLength of Edges

Aticle Related to Triangular Pyramid:

• Rectangular Pyramid
• Square Pyramid
• Pentagonal Pyramid
• Hexagonal Pyramid
• Volume of a Pyramid Formula
• Surface Area of a Pyramid Formula

Example 1: Determine the volume of a triangular pyramid whose base area and height are 50 cm² and 12 cm, respectively.

Solution:

Given data,

• Area of the triangular base = 100 cm²
• Height of the pyramid = 12 cm

We know that,

Volume of a triangular pyramid (V) = 1/3 × Area of triangular base × Height

V = 1/3 × 50 × 12 = 200 cm³

Hence, the volume of the given triangular pyramid is 200 cm³.

Example 2: Find the total surface area of a regular triangular pyramid when the length of each edge is 8 inches.

Solution:

Given data,

• Length of each edge of a regular triangular pyramid (a) = 8 inches

We know that,

Total surface area of a regular triangular pyramid = √3a²

⇒ TSA = √3 × 8²

= 64√3 = 110.851 sq. in

Hence, the total surface area of a regular triangular pyramid is 110.851 sq. in.

Example 3: Determine the volume of a regular triangular pyramid when the length of the edge is 10 cm.

Solution:

Given data,

• Length of each edge of a regular triangular pyramid (a) = 10 cm

We know that,

Volume of a regular triangular pyramid = a³/6√2

⇒ V = (10)³/6√2

= 1000/6√2 = 117.85 cm³

Hence, the volume of a regular triangular pyramid is 117.85 cu. cm.

Example 4: Find the slant height of the triangular pyramid if its lateral surface area is 600 sq. in. and the perimeter of the base is 60 inches.

Solution:

Given data,

• Lateral surface area = 600 sq. in
• Perimeter of the base = 60 inches

We know that,

Lateral surface area = ½ × perimeter × slant height

600 = ½ × 60 × l

l = 600/30 = 20 inches

Hence, the slant height of the given pyramid is 20 inches.

Example 5: Determine the total surface area of a triangular pyramid whose base area is 28 sq. cm, the perimeter of the triangle is 18 cm, and the slant height of the pyramid is 20 cm.

Solution:

Given data,

• Area of Triangular Base = 28 cm²
• Slant height (l) = 20 cm
• Perimeter (P) = 18 cm

We know that,

Total surface area (TSA) of a triangular pyramid = ½ × perimeter × slant height + Base area

⇒ TSA = ½ × 18 × 20 + 28

= 180 + 28 = 208 sq. cm

Hence, the total surface area of the given pyramid is 208 sq. cm.

Q1. Given a triangular pyramid with a base area of 15 square units and a height of 10 units, what is the volume of the pyramid?

Q2. Given a regular triangular pyramid with each edge of the equilateral triangular base measuring 6 units, what is the total surface area of the pyramid?

Q3. Given a regular triangular pyramid with each edge of the equilateral triangular base measuring 4 units and a height of 5 units, what are the volume and total surface area of the pyramid?

Q4. If the side lengths of the base of a triangular pyramid are 3 units, 4 units, and 5 units, and the height of the pyramid is 12 units, what is the volume of the pyramid?

Q5. For a triangular pyramid with a base in the shape of a right triangle with legs of 3 units and 4 units, and hypotenuse of 5 units, what is the total surface area if the height of the pyramid from the base to the apex is 10 units?

What is Definition of a Triangular Pyramid?

A triangular pyramid is a geometric shape that has a triangular base and three triangular faces, having a common vertex.

How Many Faces and Vertices do a Triangular Pyramid Have?

Triangular pyramid has four faces and  four vertices. One vertex is common to all three faces of the pyramid.

What is Basic Formula for a Pyramid?

Basic formulas of a pyramid are:

• LSA = ½ × Perimeter × Slant Height
• TSA = ½ ×  P ×  l + ½ bh
• V = 1/3 × AH

What are Types of Triangular Pyramids?

There are three types of triangular pyramids which are

• Regular Triangular Pyramid
• Irregular Triangular Pyramid
• Right-Angled Triangular Pyramid

What is Formula for Triangles?

The formula for area of triangle is:

• (Area)A = 1/2 × b × h