A triangular pyramid is a geometric solid with a triangular base, and all three lateralfaces 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.

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A three-dimensional shape with all its four faces as triangles is known as a triangular pyramid.

1.What isTriangular Pyramid?
2.Types of Triangular Pyramid
3.Propertiesof a Triangular Pyramid
4. Triangular Pyramid Formulas
5. Solved Examples onTriangular Pyramid
6.Practice Questions on Triangular Pyramid
7.FAQs on Triangular Pyramid

What isTriangular 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. Thebase is a right-angle triangle in a right triangular pyramid, while other faces areisosceles triangles.

Triangular Pyramid Nets

The net patternis different for different types of solids.Nets are usefultofind the surface area of ​​solids. A triangular pyramid netis a pattern that forms when its surface is laid flat, showing each triangular facet of a shape. The triangular pyramid netconsists 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 differentnets of a triangular pyramidshown below.Copy this on thesheet of paper. Cut it along the edge and fold it as shown in the picture below. The folded paper forms atriangular 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 pyramidhas equilateral triangles as its faces. Since it is made of equilateral triangles, all itsinternal angles will measure 60°.


Irregular Triangular Pyramid

An irregular triangular pyramidalso has triangular faces, but they are not equilateral. The internalangles in each plane add up to 180° as theyare triangular. Unless a triangular pyramidis specificallymentioned asirregular,all triangular pyramidsare assumed to beregular triangular pyramids.


Right Triangular Pyramid

The right triangular pyramid (a three-dimensional figure) has a right-angle triangle base and the apex aligned above the center of the base. It has1 base, 6 edges, 3 faces, and 4 vertices.

Important Notes

A triangular pyramidhas 4 faces, 6 edges, and 4 vertices.All four faces are triangular in shape.

Propertiesof 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 Propertiesof a Triangular Pyramid are:

It has 4 faces, 6 edges, and 4 vertices.At each of its vertex, 3 edges meet.A triangular pyramidhas no parallel faces.Triangular Pyramidsare found asregular, irregular, and right-angled.

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 pyramidwith 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 PyramidVolume = 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 pyramidmade of equilateral triangles of side a.


Regular Triangular Pyramid Volume = a3/6√2

Regular Triangular PyramidSurface Area(Total) = √3a2

Right Triangular Pyramid Formulas

Surface AreaofaRight 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:

Rohan hasa tent that is shaped likean irregular triangular pyramid. The volume of the tent is v cubic cm, and the height is h cm. What would be the areaof the base of histent?

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Example 1: Sid got to know that two triangular pyramids were congruent.He startedobserving themfor their congruency. While he placed the base of both the triangles in a position to see if theyoverlap, the two congruent triangular pyramidsstuck together along its base andformed a triangular bipyramid. How many faces, edges, and vertices does this bipyramid have?


Solution: If we openup theabove image to see the net of the triangular bipyramid,we can observe this:


There are6 triangular faces, 9 edges, and 5 vertices. ∴ Triangular bipyramid has 6 triangular faces, 9 edges, and 5 vertices.

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Example 2: Find the volume of a regular triangular pyramidwith a side length measuring5 units. (Round off the answer to 2 decimal places)

Solution: We know that for a triangular pyramidwhose side is a volume is:a3/6√2. Substituting a = 5, we get

Volume = 53/6√2

= 125/8.485


∴The volume of thetriangular pyramid is 14.73 units3

Example 3: Each edge of a regular triangular pyramidis of length 6 units. Find its total surface area.

Solution: The total surface area of a regular triangular pyramidof side ais:√3a2. Substituting a= 6, we get,

TSA =√3 × 62= √3 × 6 × 6

= 62.35

∴ Total Surface Area = 62.35 units2

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 pyramidis 60 units. Find the surface area of one of its faces."

Solution: We know that atriangular pyramidhas 6 edges. And it's given to be a regular triangular pyramid. Therefore, the length of each edge is:60/6 = 10units. The surface area of one face of the triangular pyramid: