Triangular Pyramid

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Pyramid

The pyramid is a polyhedron developed by a polygonal base and a point called the apex in geometry. Each base edge & apex form a triangle called a side face. This is a conic solid with a polygonal base. The pyramid with its n-sided base has n+1 vertices, n+1 faces, and 2n edges. 

Types of Pyramids

There are various kinds of pyramids whose names are based on the shapes of their bases.

• Triangular Pyramid

• Square Pyramid

• Pentagonal Pyramid

• Right Pyramid

• Oblique Pyramid

Triangular Pyramid

A triangular pyramid is a kind of pyramid with a triangular base. Vertices are essentially corners in geometry. All triangular pyramids, either regular or irregular, have four vertices. Triangular pyramids are made up entirely of triangles. They have 6 edges, 3 are along the base and 3 are extending up from the base. When six edges are of the same length, all the triangles are equilateral, and the pyramid would be called the regular tetrahedron. A Rubik's triangle is an example of a triangular pyramid.

Parts of a Triangular Pyramid

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A triangular pyramid has three main components. First is the base, which is, of course, a triangle. Triangular pyramid faces, which are three triangles, are next. The last one is the apex, the point at the top of which all the faces meet. There are also a few major measurements: height, base length, length of the apothem, and height of the slant. The height of the slant, base length, and length of the apothem is shown in blue.  Height is the perpendicular line that goes from the point of the triangle to the base's midpoint.

Properties of Triangular Pyramid

• It's got four faces. 

• The three sides are triangles. 

• The base is a triangle too. 

• It's got four vertices. 

• It'll be having six edges and also a tetrahedron.

Note: When not specified, the pyramid is usually assumed to be a regular square pyramid, like the physical pyramid structures. A triangular based pyramid is more often referred to as a tetrahedron. The base of the pyramid may be based on a variety of triangles. When all faces are equilateral triangles or triangles whose edges are equal in length, the pyramid is named the regular tetrahedron. When the triangles have edges of different lengths, the pyramid is an irregular tetrahedron.

Different Triangular Pyramid Types

• Regular Triangular Pyramid

• Non-Regular Triangular Pyramid

• Right Triangular Pyramid

Formulas of Triangular Pyramid

Let us now discuss the formula to find the Area and volume of a triangular pyramid. These formulas are used to solve the problems based on triangular pyramids.

Volume of a Triangular Pyramid

We can find Volume of a Triangular Pyramid by Multiplying the Area of the triangular base and the pyramid's height and then divide the number by three.

 => Volume = ⅓ × Base Area × Height

Surface Area of Triangular Pyramid:

Add the Area of the base and the Area of all sides together to calculate the surface area of a triangular-based pyramid. This calculation is simple for regular tetrahedra. Find the base's length and the height of one of the triangles. Multiply those measurements individually and divide this number by two. This will be Area of one of the triangles. Now, to account for all the triangular faces of the pyramid, multiply this Area by four. For irregular tetrahedra, using the Area theorem, find the Area of each triangle individually. Then, add together all the regions.

=> Surface Area = (Base area) + ½ × Perimeter × (Slant length)

Example Problems

Example 1: Find a triangular pyramid's surface area with a base area of 28cm², a perimeter of 20 cm, a slant length of 5 cm.

Ans: We know that, 

Surface Area = (Base area) + ½ × Perimeter × (Slant length)

Let us substitute the given Area, perimeter and Slant height in the formula,

We get, 

28 + ½ × 20 × 5

28 + 50

78 cm2

Example 2: If the base area is 28cm², height is 4.5 cm then find the volume of a triangular pyramid.

Ans: We know that, 

Volume = ⅓ × Base Area × Height

Let us substitute the given Base area, Slant height in the formula,

We get,

⅓ × 28 × 4.5

⅓ × 126

42 cm3

Triangular Prism

A polyhedron (three-dimensional shape), consisting of two triangular bases and three rectangular sides, is a triangular prism. The two faces here are parallel and congruent to each other, like other prisms. It has a total of 5 sides, 6 vertices, and 9 edges.

Properties of a Triangular Prism

• It has two bases with triangles and three sides with rectangles. 

• Instead of a rectangle, if the triangular bases are equilateral and the other faces are circular, then the triangular prism is semiregular. 

How Many Edges Has a Triangular Prism? 

A triangular prism has 9 edges, 5 sides, and 6 vertices in total (which are joined by the rectangular faces). 

Activity-1: Construction of a Triangular Pyramid

Follow this procedure to make a Triangular Pyramid on your own. We will need 4 triangles that connect at the edges to create a closed 3-D shape or six edge pieces and 4 corner pieces to make a frame in order to build a triangular pyramid.

1. First, let’s take a sheet of paper. 

2. We’ll make similar lines on the sheet mentioned in the paper below. 

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3. Then we’ll cut the sheet in the edges and fold it as guided in the figure shown below to get a tetrahedral shape.  

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4. The folded paper will form a Triangular Pyramid or this can also be called a tetrahedron.

Activity-2: How to Draw a Triangular Pyramid

1. Draw a triangle (ABC). 

2. Find it's middle point (P) (connect all corners with the middle of the side that faces). 

3. Draw a straight line upwards from the middle point (AP). 

4. Connect each corner of the triangle to point P.

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Conclusion

We learned the interesting idea of the Triangular Pyramid in this lesson. We compared the number of faces, corners, and vertices, measured surface area, and volumes, and learned how to create a pyramid. We prepared this lesson so that it is relatable and easy to understand, but will stick with the students forever as well. We made sure that in young minds, this creatively generates a fresh notion.