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A triangular pyramid is a geometric solid with a triangular base, and all three lateral faces are also triangles with a common vertex. The tetrahedron is a triangular pyramid with equilateral triangles on each face. Four triangles form a triangular pyramid. Triangular pyramids are regular, irregular, and right-angled.
A three-dimensional shape with all its four faces as triangles is known as a triangular pyramid.
|1.||What is a Triangular Pyramid?|
|2.||Triangular Pyramid Formulas|
|3.||FAQs on Triangular Pyramid|
What is a Triangular Pyramid?
A triangular pyramid is a 3D shape in which all the faces are triangles. It is a pyramid with a triangular base connected by four triangular faces where 3 faces meet at one vertex. If it is a right triangular pyramid, the base is a right-angled triangle while the other faces are isosceles triangles.
Triangular Pyramid Nets
The net pattern is different for different types of solids. Nets are useful to find the surface area of solids. A triangular pyramid net is a pattern that forms when its surface is laid flat, showing each triangular face in the two-dimensional (2D) form. The triangular pyramid net consists of four triangles.
Let us do a small activity to learn more about the net of a triangular pyramid. Take a sheet of paper. You can observe two different nets of a triangular pyramid shown below. Copy this on the sheet of paper. Cut it along the edge and fold it as shown in the picture below. The folded paper forms a triangular pyramid.
Types of Triangular Pyramid
Like any other geometrical figure, triangular pyramids can also be classified into regular and irregular pyramids. The different types of triangular pyramids are explained below.
Regular Triangular Pyramid
A regular triangular pyramid has equilateral triangles as its faces. Since it is made of equilateral triangles, all its internal angles measure 60°.
Irregular Triangular Pyramid
An irregular triangular pyramid also has triangular faces, but they are not equilateral. The internal angles in each plane add up to 180° as they are triangular. Unless a triangular pyramid is specifically mentioned as irregular, all triangular pyramids are assumed to be regular triangular pyramids.
Right Triangular Pyramid
A right triangular pyramid (a three-dimensional figure) has the base as a right-angled triangle and the apex is aligned above the center of the base. It has 1 right-angled base, 6 edges, 3 triangular faces, and 4 vertices.
Properties of a Triangular Pyramid
The properties of a triangular pyramid help us to identify a pyramid from a given set of figures quickly and easily. The different properties of a triangular pyramid are:
- A triangular pyramid has 4 triangular faces, 6 edges, and 4 vertices.
- 3 edges meet at each of its vertex.
- A triangular pyramid has no parallel faces.
- A regular triangular pyramid has equilateral triangles for all its faces. It has 6 planes of symmetry.
- Triangular pyramids can be regular, irregular, and right-angled.
Triangular Pyramid Formulas
There are various formulas that are used to calculate the volume and surface area of triangular pyramids. Observe the following figure to relate to the formulas given below:
- The volume of a triangular pyramid is calculated with the formula, Volume of Triangular Pyramid = 1/3 × Base Area × Height; where we multiply the area of the triangular base by the height of the pyramid (measured from base to top) and then divide that product by 3 as shown in the formula.
- The total surface area of a triangular pyramid is calculated with the formula, Total Surface area of a Triangular Pyramid = Base Area + 1/2 (Perimeter of the base × Slant Height); where 'slant height' is the distance from its triangular face along the center of the face to the apex.
- Now consider a regular triangular pyramid made of equilateral triangles of side 'a'. The two main formulas of a regular triangular pyramid are: Volume of a Regular Triangular Pyramid = a3/6√2 and Total Surface Area of a Regular Triangular Pyramid = √3a2
Tips on Triangular Pyramid
- A triangular pyramid has 4 faces, 6 edges, and 4 vertices. All four faces are triangular in shape.
- The tetrahedron is a triangular pyramid having congruent equilateral triangles for each of its faces.
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Triangular Pyramid Examples
Example 1: If two congruent triangular pyramids are stuck together along their base they form a triangular bipyramid. How many faces, edges, and vertices does this bipyramid have?
Solution: This triangular bipyramid has 6 triangular faces, 9 edges, and 5 vertices.
Example 2: Find the volume of a regular triangular pyramid with a side length measuring 5 units. (Round off the answer to 2 decimal places)
Solution: We know that for a triangular pyramid whose side is a volume is: a3/6√2. Substituting a = 5, we get
Volume = 53/6√2
∴ The volume of the triangular pyramid is 14.73 units3
Example 3: Each edge of a regular triangular pyramid is of length 6 units. Find its total surface area.
Solution: The total surface area of a regular triangular pyramid of side a is: √3a2. Substituting a = 6, we get,
TSA = √3 × 62 = √3 × 6 × 6
∴ Total Surface Area = 62.35 units2
Practice Questions on Triangular Pyramid
FAQs on Triangular Pyramid
What is a Triangular Pyramid in Math?
A triangular pyramid is a 3D shape in which all the faces are triangles. It is a geometric solid with a triangular base, and all three lateral faces are also triangles with a common vertex.
What is the Volume of a Triangular Pyramid?
The volume of a triangular pyramid is the space inside the pyramid in a three-dimensional (3D) plane. The formula for the volume of a triangular pyramid is expressed as, Volume of a Triangular Pyramid = 1/3 × Base Area × Height, where, base area is the area of the triangular base and height is the height of the pyramid measured from the base to the top vertex.
What is the Surface Area of a Triangular Pyramid?
The surface area of a triangular pyramid is the sum of the areas of all of the faces of the pyramid. The formula that is used to find the surface area of a triangular pyramid is, Total Surface area of Triangular Pyramid = Base Area + 1/2 (Perimeter of the base × Slant Height); where the slant height is the distance from the triangular face along the center of the face to the apex.
What is the Base of a Triangular Pyramid?
The base of a triangular pyramid is also a triangle which means there are 4 triangular faces in a triangular pyramid.
Give an Example of a Triangular Pyramid.
A common example of a triangular pyramid is the Pyramix or the Rubik's triangle which has 1 triangular base and 3 other triangular faces.
How many Faces, Edges and Vertices does a Triangular Pyramid have?
A triangular pyramid has 4 triangular faces, 6 edges (sides) and 4 vertices (corners). The 4 triangular faces include the base of the pyramid which is also a triangle.
How Many Triangles Form a Triangular Pyramid?
Four triangles form a triangular pyramid out of those, one is the base and the other three triangles are the three lateral faces of the pyramid.
How Many Vertices Does a Triangular Pyramid have?
A triangular pyramid has 4 vertices. The three edges of the pyramid intersect at each vertex. Out of the 4 vertices, 3 vertices are at the base of the triangular pyramid and one vertex is at the top.
What are the Different Types of Triangular Pyramids?
There are three types of triangular pyramids and those are - regular, irregular, and right-angled triangular pyramids.
Is a Triangular Pyramid made of Equilateral Triangles?
All triangular pyramids are not made of equilateral triangles. When a triangular pyramid is made up of equilateral triangles, it is called a regular triangular pyramid. It has four faces, including the base.