Triangular Pyramid
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 rightangled.
A threedimensional shape with all its four faces as triangles is known as a triangular pyramid.
What is Triangular Pyramid?
A triangular pyramid is a 3D shape, all of the faces of which are made in the form of triangles. A triangular pyramid is a pyramid with a triangular base and bounded by four triangular faces where 3 faces meet at one vertex. The base is a rightangle triangle in a right triangular pyramid, while 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 facet of a shape. The triangular pyramid net consists of four triangles. The base of the pyramid is a triangle; the side faces are also triangles.
Let us do a small activity. 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 will 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
The right triangular pyramid (a threedimensional figure) has a rightangle triangle base and the apex aligned above the center of the base. It has 1 base, 6 edges, 3 faces, and 4 vertices.
Important Notes
 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.
Properties of a Triangular Pyramid
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:
 It has 4 faces, 6 edges, and 4 vertices.
 At each of its vertex, 3 edges meet.
 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 are found as regular, irregular, and rightangled.
Triangular Pyramid Formulas
There are various formulas to calculate the volume, surface area, and perimeter of triangular pyramids. Those formulae are given below:
To find the volume of a pyramid with a triangular base, multiply the area of the triangular base by the height of the pyramid (measured from base to top). Then divide that product by three.
Triangular Pyramid Volume = 1/3 × Base Area × Height
The slant height of a triangular pyramid is the distance from its triangular base along the center of the face to the apex. To determine the surface area of a pyramid with a triangular base, add the area of the base and the area of all sides.
Triangular Pyramid Surface Area(Total) = Base Area + 1/2 (Perimeter × Slant Height)
Now consider a regular triangular pyramid made of equilateral triangles of side a.
Regular Triangular Pyramid Volume = a^{3}/6√2
Regular Triangular Pyramid Surface Area(Total) = √3a^{2}
Right Triangular Pyramid Formulas
Surface Area of a Right Triangular Pyramid (\(A_{s}\)) = 1/2 (\(h_{b}\) × a) + 3/2 (a × \(h_{s}\))
The volume of a Right Triangular Pyramid (V) = 1/6 × \(h_{b}\) × a × h = 1/3 × \(A_{b}\) × h
Where \(A_{s}\) = Surface Area, \(A_{b}\) = Base Area, V= Volume, a = Edge, h = Height, \(h_{b}\) = Height Base, and \(h_{s}\) = Height Side.
Challenging Questions:
 A new shape is formed by aligning the face of a triangular pyramid exactly over one triangular face of the square pyramid. How many vertices, edges, and faces will the new shape have?
 Rohan has a tent that is shaped like an irregular triangular pyramid. The volume of the tent is v cubic cm, and the height is h cm. What would be the area of the base of his tent?
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Solved Examples on Triangular Pyramid

Example 1: Sid got to know that two triangular pyramids were congruent. He started observing them for their congruency. While he placed the base of both the triangles in a position to see if they overlap, the two congruent triangular pyramids stuck together along its base and formed a triangular bipyramid. How many faces, edges, and vertices does this bipyramid have?
Solution: If we open up the above image to see the net of the triangular bipyramid, we can observe this:
There are 6 triangular faces, 9 edges, and 5 vertices. ∴ 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: a^{3}/6√2. Substituting a = 5, we get
Volume = 5^{3}/6√2
= 125/8.485
≈14.73
∴ The volume of the triangular pyramid is 14.73 units^{3}

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: √3a^{2}. Substituting a = 6, we get,
TSA = √3 × 6^{2} = √3 × 6 × 6
= 62.35
∴ Total Surface Area = 62.35 units^{2}

Example 4: While solving questions about the triangular pyramid, Syna got stuck. Let's help her out to reach the final answer. Here's the question: "The sum of the length of the edges of a regular triangular pyramid is 60 units. Find the surface area of one of its faces."
Solution: We know that a triangular pyramid has 6 edges. And it's given to be a regular triangular pyramid. Therefore, the length of each edge is: 60/6 = 10 units. The surface area of one face of the triangular pyramid:
Area of a face = √3/4 × a^{2}
Substituting a = 10, we get:
Area of a face = √3/4 × 10^{2} = √3/4 × 10 × 10
= 25√3
= 43.30
∴ Surface Area of one of its faces = 43.30 units^{2}
Practice Questions on Triangular Pyramid
FAQs on Triangular Pyramid
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. At each vertex, three edges intersect. Out of 4 vertices, three 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 pyramid and those are  regular, irregular, and rightangled triangular pyramid.
Is a Pyramid a Triangular Prism?
A prism is a polyhedron with parallel top and bottom bases and rectangular side faces. A pyramid has one base and triangular side faces, which meet at a central vertex point. A triangular pyramid is a geometric solid with a triangular base, and all other faces are also triangles with a common vertex. So, a pyramid is not a triangular prism.
What is a 6 Sided Pyramid Called?
In geometry, a hexagonal pyramid is a pyramid with a hexagonal base on which six isosceles triangular faces rise, which meet at one point (apex).
Why is a Pyramid a Triangle?
Each side of a pyramid (each base edge and the apex) forms a triangle. A trianglebased pyramid has four triangular sides.
Is a Pyramid made of Equilateral Triangles?
When a pyramid is made up of equilateral triangles, it is called a regular triangular pyramid. It has four faces, including the base.