# How to Find the Volume and Surface Area of a Triangular Pyramid?

In this step-by-step guide, you learn how to use formulas to find the volume and surface area of a triangular pyramid.

A three-dimensional shape whose four sides are triangular is known as a triangular pyramid. Triangular pyramids are irregular, regular, and right-angled.

## Step by step guide to finding the volume and surface area ofthe triangular pyramid

A triangular pyramid is a three-dimensional shape in which all faces are triangular. It is a pyramid with a triangular base connected by four triangular faces in which $$3$$ faces meet at a vertex. If it is a right triangular pyramid, the base is a right-angled triangle and the other sides of the triangle are isosceles.

### Types of the triangular pyramid

Like any other geometric shape, triangular pyramids can be classified into regular and irregular pyramids.

#### Regular triangular pyramid

A regular triangular pyramid has equilateral triangles. Since it is made of equilateral triangles, its interior angles are $$60$$ degrees.

#### Irregular triangular pyramid

Irregular triangular pyramids also have triangular faces, but they are not equilateral. The internal angles in each plane add up to $$180°$$ due to the triangle. Unless a triangular pyramid is mentioned explicitly as irregular, all triangular pyramids are assumed to be regular triangular pyramids.

#### Right triangular pyramid

The right triangular pyramid has the base of the right triangle and its apex is located 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

A triangular pyramid features help us identify a pyramid from a set of specific shapes 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 vertexes.
• 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 used to calculate the volume and area of triangular pyramids. See the following figure to relate to the formulas given below:

We can calculate the volume of a triangular pyramid with this formula:

$$\color{blue}{Volume\:of\:Triangular\:Pyramid\:=\:\frac{1}{3}\:×\:Base\:Area\:×\:Height}$$

Where we multiply the area of the triangular base by the height of the pyramid (measured from the base to up) and then divide the result by $$3$$ according to the formula.

We can calculate the the total area of a triangular pyramid by this formula:

$$\color{blue}{Total\:Surface\:area\:of\:a\:Triangular\:Pyramid\:=\:Base\:Area\:+\:\frac{1}{2}\:\left(Perimeter\:of\:the\:base\:×\:Slant\:Height\right)}$$

where slant height is the distance from its triangular face along the center of the face to the apex.

### Finding the Volume and Surface Area of the Triangular Pyramid –Example 1:

Find the volume of a triangular pyramid having a base area of $$15\space cm^2$$ and a height of $$6\space cm$$.

Solution:

Using this formula to find the volume of a triangular pyramid:

$$Volume=\frac{1}{3}×\:Base\:area\:×\:Height$$

$$= \frac{1}{3}× 15 × 6$$

$$=\frac{90}{3}$$

$$= 30\space cm^3$$

## Exercises for Findingthe Volume and Surface Area of the Triangular Pyramid

### Find the volume and surface area for eachtriangular pyramid.

• $$\color{blue}{V= 99.2\space ft^3, A= 145.3\space ft^2}$$
• $$\color{blue}{V= 46.5 \space cm^3, A=95.8\space cm^2}$$
• $$\color{blue}{V=43.4 \space cm^3, A=85.7\space cm^2}$$

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