Triangular Pyramid Formula

A triangular pyramid has a triangle-shaped base and all three triangular faces meet at the apex. There is a special case of a triangular pyramid called a tetrahedron, it has equilateral triangles for each of the faces. The triangular pyramid formula consists of both the volume and the surface area of the triangular pyramid that calculates the three triangular-shaped sides, the height, and the slant height. The following figure shows how a triangular pyramid looks like:

What is Triangular Pyramid Formula?

A pyramid with a triangle-shaped base whose three triangular faces meet at the apex. The triangular pyramid formula included both the volume and surface area of the pyramid. The triangular pyramid volume formula calculates the base area and the height whereas the surface area of the triangular pyramid calculates the base area, perimeter, and slant height. Formulas for volume and surface area of the triangular pyramid are given below that are used in the triangular pyramid formula: 

Volume= 1/3 × Base area ×Height

Surface Area = Base area +1/2(perimeter × slant height)

Triangular Pyramid Formula 

Formulas for volume and surface area of the triangular pyramid are: 

Volume= 1/3 × Base area ×Height

Surface Area = Base area +1/2(perimeter × slant height)

Examples Using Triangular Pyramid Formula

Example 1: Find the volume of a triangular pyramid having a base area of 10 cm² and a height of 5 cm.

Solution:

Given: base area = 10 cm², height = 5 cm.

Using the formula for the volume of a triangular pyramid

Volume =1/3 × Base area × Height

= 1/3 × 10 × 5

= 16.67 cm³

Therefore, the volume of the triangular pyramid is 16.67 cm³

Example 2: A triangular pyramid has a base area of 15 units² and a sum of the lengths of the edges 60 units. Calculate the surface area of the triangular pyramid if the slant height is 20 units.

Solution:
Given: Base area =15 units², perimeter = 60 units.

Using the formula for surface area of triangular pyramid

Surface Area = Base area + 1/2(perimeter × slant height)

= 15 + 1/2(60 × 20)
= 15 + 600
= 615 unit²

Therefore, the surface area of the triangular pyramid is 615 unit²

Example 3: Find the surface area of a triangular pyramid whose area of the base triangles is 24 square units, the perimeter of the triangle is 12 units, and the slant height of the pyramid is 18.

Solution:

The area of the base triangles = 24 squared units.

The perimeter of the triangle =12 units.

The slant height of the pyramid =18 units.

Using the formula for the surface area of a triangular pyramid

The surface area of a triangular pyramid = Base Area+ 1/2(Perimeter × Slant Height)

= 24 + 1/2 (12 × 18)

= 132 unit²

Therefore, the surface area of a triangular pyramid 132 unit²

FAQs on Triangular Pyramid Formula

What is Meant by Triangular Pyramid Formula?

A triangular pyramid has a triangle-shaped base and all three triangular faces meet at the apex. The triangular pyramid formula included both the volume and surface area of the pyramid. The triangular pyramid volume formula calculates the base area and the height whereas the surface area of the triangular pyramid calculates the base area, perimeter, and slant height. Formulas for volume and surface area of the triangular pyramid are given below that are used in the triangular pyramid formula: 

Volume= 1/3 × Base area × Height

Surface Area = Base area +1/2(perimeter × slant height)

How Do You Find the Surface Area of a Triangular Pyramid?

The formula for the surface area of a pyramid is calculated by adding up the area of all triangular faces of a pyramid. which is Base area +1/2(perimeter × slant height). The dimensions required to find the surface area of a triangular pyramid are the side, height, and slant height.

How Do You Find the Volume of a Triangular Pyramid?

The formula for calculating the volume of a triangular pyramid is Volume= 1/3 × Base area × Height. The dimensions required to find the surface area of a triangular pyramid are the side, height, and slant height.

Using the Triangular Pyramid Formula, Find the Volume with a Base Area of 15 cm² and a height of 4 cm.

Given: base area = 15 cm², height = 4 cm.

Using the formula for the volume of a triangular pyramid

Volume =1/3 × Base area × Height

= 1/3 × 15 × 4

= 20 cm³

Therefore, the volume of the triangular pyramid is 20 cm³

Surface Area of Triangular Pyramid Formula

The surface area of a triangular pyramid is the total area of all faces of a triangular pyramid. Basically, a triangular pyramid has a triangular base and is bounded by three lateral triangular faces that meet at one vertex. A triangular pyramid has all faces as triangles. This pyramid has 4 faces, 6 edges, and 4 corners or vertices. Few types of the triangular pyramid are given below:

• Regular triangular pyramid – all faces are equilateral triangles and are known as tetrahedrons.
• Right triangular pyramid – the base is an equilateral triangle while other faces are isosceles triangles.
• An irregular triangular pyramid – a scalene or isosceles triangle forms the base.

What Is The Surface Area of a Triangular Pyramid?

The surface area of any three-dimensional geometrical shape is the sum of the areas of all of the faces or surfaces of that enclosed solid. A triangular pyramid has four triangular faces. Thus, The formula for calculating the surface area of a triangular pyramid involves the area of the base, the perimeter of the base, and the slant height of any side of the pyramid. The surface area is always measured in square units like cm², m², ft², or cubits². The surface area of a triangular pyramid is Base Area+1/2(Perimeter×Slant Height).

Surface Area of Triangular Pyramid Formula

The Formula for the surface area of a triangular pyramid is calculated by adding up the area of all triangular faces of a pyramid. The surface area of a right triangular pyramid formula is Base Area+1/2(Perimeter×Slant Height)

.After putting the values we get an expression of the surface area of the triangular pyramid formula as 1⁄2(a × b) + 3⁄2(b × s).

Where,

• b is the side of the triangle pyramid.
• a is the height of the base triangle
• s is the slant height of a triangular pyramid.
• 1⁄2(a × b) is the base area of the triangular faces.
• 3⁄2(b × s) is the product of the perimeter and slant height of a pyramid.

How to Calculate the Surface Area of Triangular Pyramids?

The surface area of a triangular pyramid can be calculated by representing the 3D shape into a 2D net, to make the shapes easier to see. After expanding the 3D shape into 2D shape we will get four triangles.
The following steps are used to calculate the surface area of a triangular pyramid:

• Step 1: Find the area of the base triangles: The area of the base triangles is (1/2 × base of the triangle × height of the triangle) which becomes base × height.
• Step 2: Find the perimeter of triangular faces: The perimeter of a triangle is the sum of all sides of a triangle which is (side)1

+ (side)2 + (side)3. Step 3: Find the slant height of triangular faces: The slant height of a triangular pyramid is generally represented by ‘s’. Step 4: Add all the areas together. Thus, the surface area of a triangular pyramid formula is 1⁄2(a × b) + 3⁄2(b × s) in squared units.

Lateral Surface Area of Triangular Pyramid

The lateral surface area is the area of the non-base faces or we can say that only the lateral surface area of any object is calculated by removing the base area. The lateral area of a triangular pyramid can be calculated by removing the base area of a triangle from the product of the perimeter of the base and the slant height of a pyramid.

Thus, the lateral surface area of a right triangular pyramid is 1⁄2(perimeter of the base × slant height) which further becomes 3⁄2(side × slant height).

Where,

• b is the side of a pyramid.
• s is the slant height of the base.

Examples on Surface Area of Triangular Pyramid Formula

Example 2: Find the surface area of a triangular pyramid whose area of the base triangle is 24 units², the perimeter is 12 units, and the slant height of the pyramid is 18.

Solution

The surface area of a triangular pyramid of side a is

Surface Area = 1⁄2(a × b) + 3⁄2(b × s)

The surface area of a triangular pyramid = $\begin{array}{c}\text{Base Area}+\frac{\mathrm{1/}}{2}\text{(Perimeter}\times \text{Slant Height})\end{array}$

Putting the values in the formula,

The surface area of a triangular pyramid = $\begin{array}{c}\text{24}+\frac{\mathrm{1/}}{2}\text{(12}\times \text{18})\end{array}$

= 132 square units.

Answer: The surface area of a triangular pyramid is 96 units².

How to use our triangular pyramid volume calculator?

A triangular pyramid is a solid object formed by connecting a triangular base to a point, called the apex. This creates four faces, each of which is a triangle. If you can rotate the pyramid, each face can play the role of the pyramid’s base. The segment that is perpendicular to the base and runs through the apex is known as the height of the pyramid.

Here the dark blue triangle is the base of the pyramid, and the dotted black segment is the height:

To use our triangular pyramid volume calculator, follow these steps:

1. Do you know the area of the pyramid’s base?

• If so, enter it into the calculator.
• If not, check what data about the base you do know: choose the appropriate option in our calculator and enter your data. The base area will be calculated for you.

2. Enter the pyramid’s height.
3. The triangular pyramid volume calculator will then return your pyramid’s volume 🙂

Triangular pyramid volume formula

The triangular pyramid volume formula resembles the formulas used in our pyramid volume calculator and cone volume calculator to obtain those quantities. The triangular pyramid volume formula is:

V = A × H / 3

, where:

• V is the triangular pyramid volume;
• A is the area of the pyramid’s base; and
• H is the height from the base to the apex.

In words: the volume of a triangular pyramid is one-third of the product of the base area and the pyramid’s height.

Triangular Pyramid

A 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.

Triangular Pyramid Formula

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

The 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 (LSA) = ½ × perimeter × slant height

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 (TSA) of a triangular pyramid = ½ ×  P ×  l + ½ bh

where,

P is the perimeter of the base,

l is the slant height of the pyramid,

b is the base of the base triangle, and

h is the height of the pyramid

Volume of a Triangular Pyramid

The 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 the volume of the pyramid,

A is the area of the base of a pyramid, and

H is the height or altitude of a pyramid.

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

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

Where a is the length of the edges.

Practise Problem  based on Triangular Pyramid Formula

Problem 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

The height of the pyramid = 12 cm

We know that,

The 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.

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

Solution:

Given data,

The length of each edge of a regular triangular pyramid (a) = 8 inches

We know that,

The 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.

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

Solution:

Given data,

The length of each edge of a regular triangular pyramid (a) = 10 cm

We know that,

The 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.

Problem 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,

The lateral surface area = 600 sq. in

The perimeter of the base = 60 inches

We know that,

The 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.

Problem 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 the triangular base = 28 cm²

The slant height (l) = 20 cm

Perimeter (P) = 18 cm

We know that,

The 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.

Frequently Asked Questions

Question 1: What is the definition of a triangular pyramid?

Answer:

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

Question 2: How many faces and vertices do a triangular pyramid have?

Answer:

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

Question 3: What are the types of triangular pyramids?

Answer:

There are three types of triangular pyramids which are

• Regular triangular pyramids.
• Irregular triangular pyramids.
• Right-angled triangular pyramids.

How to find the Volume of a Triangular Pyramid?

A triangular pyramid is solid with a triangular base and triangles having a shared vertex on all three lateral faces. It’s a tetrahedron with equilateral triangles on each of its four faces. It has a triangle base and four triangular faces, three of which meet at one vertex. A right triangular pyramid’s base is a right-angled triangle, with isosceles triangles on the other faces. All of the faces of a regular triangular pyramid are equilateral triangles and it contains six symmetry planes.

Volume of a Triangular Pyramid

The amount of space occupied by a triangular pyramid in a 3D plane is called its volume. To put it another way, volume specifies the confined area or region of the pyramid. Knowing the base area and height of a triangular pyramid is enough to calculate its volume. Its formula equals one-third the product of base area and height. It is measured in units of cubic meters (m³). 

V = 1/3 × B × h

Where,

V is the volume,

B is the base area,

h is the height of pyramid.

If we are given a regular triangular pyramid consisting of equilateral triangles, its volume is given by the formula,

V = a³/6√2

Where,

V is the volume,

a is the side length.

How to Find the Volume of a Triangular Pyramid?

Let’s take an example to understand how we can calculate the volume of a triangular pyramid.

Example: Calculate the volume of a triangular pyramid of base area 90 sq. m and height 6 m.

Step 1: Note the base area and height of a triangular pyramid. In this example, the base area of the pyramid is 90 sq. m and height is 6 m.

Step 2: We know that the volume of a triangular pyramid is equal to 1/3 × B × h. Substitute the given value of base area and height in the formula.

Step 3: So, the volume of triangular pyramid is calculated as, V = (1/3) × 90 × 6 = 180 cu. m

Sample Problems

Problem 1: Calculate the volume of a triangular pyramid with a base area of 50 sq. m and a height of 4 m.

Solution:

We have,

B = 50

h = 4

Using the formula we get,

V = 1/3 × B × h

= (1/3) × 50 × 4

= 66.67 cu. m

Problem 2: Calculate the volume of a triangular pyramid with a base area of 120 sq. m and a height of 10 m.

Solution:

We have,

B = 50

h = 4

Using the formula we get,

V = 1/3 × B × h

= (1/3) × 120 × 10

= 400 cu. m

Problem 3: Calculate the base area of a triangular pyramid if its volume is 300 cu. m and height is 15 m.

Solution:

V = 300

h = 15

Using the formula we get,

V = 1/3 × B × h

=> B = 3V/h

=> B = 3 (300)/15

=> B = 60 sq. m

Problem 4: Calculate the base area of a triangular pyramid if its volume is 600 cu. m and height are 5 m.

Solution:

V = 600

h = 5

Using the formula we get,

V = 1/3 × B × h

=> B = 3V/h

=> B = 3 (600)/5

=> B = 360 sq. m

Problem 5: Calculate the height of a triangular pyramid if its volume is 200 cu. m and the base area is 60 sq. m.

Solution:

We have,

V = 200

B = 60

Using the formula we get,

V = 1/3 × B × h

=> h = 3V/B

=> h = 3 (200)/60

=> h = 10 m

Problem 6: Calculate the height of a triangular pyramid if its volume is 150 cu. m and the base area is 50 sq. m.

Solution:

We have,

V = 150

B = 50

Using the formula we get,

V = 1/3 × B × h

=> h = 3V/B

=> h = 3 (150)/50

=> h = 9 m

Problem 7: Calculate the volume of a regular triangular pyramid if the side length is 10 m.

Solution:

We have,

a = 10

Using the formula we get,

V = a³/6√2

= (10)³/6√2

= 117.85 cu. m

Solved Triangular Pyramid Examples

Example 1:

Determine the surface area of the triangular pyramid given in the diagram.

Solution:

Area of the base: 1/2×9×6=27

Area of the lateral face: 1/2×9×11=49.5

Find the sum of the areas of the faces.

The surface area of the triangular pyramid = Area of the base + Areas of the lateral faces. 

s = 27 + 49.5 + 49.5 + 49.5 [There are three identical lateral faces]

s = 175.5

So, the surface area is 175.5 square inches.

Example 2: Find the volume of a triangular pyramid with a base area of 28 cm2

 and a height of 4.5 cm.

Solution:
As we know, the formula for the volume of a triangular pyramid is:

Hence, the required volume of a triangular pyramid is 42 cm3

.Example 3:

A triangular bipyramid is formed when two congruent triangular pyramids are stuck together along their base. How many faces, edges, and vertices does this bipyramid have?

Solution:

There are 6 triangular faces, 9 edges, and 5 vertices in this triangular bipyramid.

Example 4:

John completes the Pyraminx in under a minute. The Pyraminx is a triangular pyramid with a base area of 27  in2

. If its height is 8 inches, determine its volume.

Solution:

As we know, the formula for the volume of a triangular pyramid is:

Examples of Triangular Pyramid

Question 1: Find the volume of a triangular pyramid when base area is

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