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A pyramid is a solid shape formed by connecting a polygon-shaped base to a common point. Pyramids are classified depending upon the shape of their base. A triangular pyramid has a triangular base, a rectangular pyramid has a rectangular base, and so on. Here, we will focus on the triangular pyramid, its types and properties, and how we can calculate its area and volume....Read MoreRead Less

A triangular pyramid is a three-dimensional shape with a triangular base and three triangular lateral faces. The lateral faces of the pyramid share a common vertex known as the apex. In other words, all three vertices of the triangular base of the pyramid are connected to the apex.

Triangular pyramids can be classified into *regular and irregular.*

**The Regular Triangular Pyramid**The base of a regular triangular pyramid is an equilateral triangle, and its apex is aligned above the center of the base. All of its internal angles measure 60 degrees.

**The Irregular Triangular Pyramid**

The triangular faces of an irregular triangular pyramid are also triangular, but they are not equilateral. The internal angles of the faces add up to 180\(^{\circ}\).

[Note: Unless a triangular pyramid is specifically described as irregular, it is assumed that all triangular pyramids are regular.]

The properties of triangular pyramids allow us to quickly and easily identify them from a set of solid shapes.

- A triangular pyramid has 6 edges, 3 triangular lateral faces, 4 vertices, and a triangular base.
- At each vertex, three edges meet.
- There are no parallel faces in a triangular pyramid.
- All of the faces of a regular triangular pyramid are equilateral triangles.
- A regular triangular pyramid has six symmetry planes.
- The height of each lateral triangle is known as the slant height of the pyramid.

The volume of a triangular pyramid is given by:

Volume = (\(\frac{1}{3} \times\) Base Area \(\times\) Height) cubic units

The height is measured from the base to the apex.

The surface area of a triangular pyramid is the sum of the area of the base and the areas of the lateral faces.

The formula for calculating the total surface area of a triangular pyramid is:

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

**Example 1:**

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

**Solution:**

Area of the base: \(\frac{1}{2} \times 9 \times 6=27\)

Area of the lateral face: \(\frac{1}{2} \times 9 \times 11=49.5\)

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 cm\(^2\) and a height of 4.5 cm.

**Solution:**

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

Volume = \(\frac{1}{3} \times\) Base Area \(\times\) Height

Now, substitute the values,

= \(\frac{1}{3} \times 28 \times 4.5\)

= \(\frac{1}{3} \times\) 126 [Simplify]

= 42 cm\(^3\) [Divide]

Hence, the required volume of a triangular pyramid is 42 cm\(^3\).

**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 in\(^2\). If its height is 8 inches, determine its volume.

**Solution:**

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

Volume = \(\frac{1}{3} \times\) Base Area \(\times\) Height

Now, substitute the values,

= \(\frac{1}{3} \times 27 \times 8\)

= \(\frac{1}{3} \times 216\) [Simplify]

= 72 in\(^3\) [Divide]

Hence, the volume of the Pyraminx is 72 in\(^3\).

Frequently Asked Questions

A tetrahedron is a polyhedron with four faces, six edges, and four vertices, each of which is a triangle. It is also known as a triangular pyramid because its base is a triangle.

An oblique pyramid is a type of pyramid whose apex is not centered over its base.

A plane of symmetry divides a shape in half, resulting in each side of the plane being a mirror image of the other side.

A rectangular pyramid is a type of pyramid with a rectangle-shaped base and triangle-shaped lateral faces. A rectangular pyramid has five faces, five vertices, and eight edges.

A pyramid is a three-dimensional polyhedron with a single polygonal base that is attached to its lateral faces, which are always triangular. A prism is also a 3D polyhedron but with two identical bases which are perpendicular to the lateral faces, and the cross-section is the same across all faces.