A triangular prism is a three-dimensional geometric figure. These prisms are polyhedra made up of two triangular bases and three lateral rectangular faces. Similar to other prisms, the two bases are parallel and congruent to each other. Triangular prisms have 5 faces, 6 vertices, and 9 edges. The edges and vertices are connected to each other through three side rectangles. These prisms are pentahedra that have nine different geometric nets.

Here, we will learn more about the characteristics of triangular prisms and use diagrams to illustrate the concepts.

## Definition of a triangular prism

A triangular prism is a type of prism that has two bases and three lateral faces. The lateral sides have a rectangular shape and the bases have a triangular shape. In total, these prisms have five faces, nine vertices, and six edges.

The lateral faces and bases of the triangular prism may or may not be congruent depending on whether the base is an equilateral triangle. The edges of the prism join the corresponding side faces. The edges of the triangles are parallel to each other.

## Fundamental characteristics of a triangular prism

The following are some of the most important characteristics of triangular prisms:

- They have a total of 9 edges.
- They have a total of 5 faces.
- They have a total of 6 vertices.
- They have two triangular faces, called bases, and three rectangular faces, called lateral faces.
- The bases are parallel and congruent with each other.
- If the bases are equilateral triangles, the lateral faces are equal to each other.

## Important triangular prism formulas

Triangular prisms are three-dimensional figures, so their most important properties are volume and surface area.

### Formula for the volume of a triangular prism

The volume of a prism is found by multiplying the area of its base by the length of its height. The bases are triangular and we know that the area of any triangle is equal to one-half of the base times the height of the triangle. Therefore, we have the formula:

$latex V=\frac{1}{2}bah$ |

where *b* represents the length of the triangle’s base, *a* represents the height of the triangle, and *h* represents the height of the prism.

### Formula for the surface area of a triangular prism

We can find the surface area by adding the areas of all the faces of the prism. We have two triangular faces that are equal, so the area of both faces is *ba*. The area of the lateral faces is equal to the length of the rectangle times its height.

If we have a prism with an equilateral triangle base, the surface area is:

$latex A_{s}=ba+3bh$ |

where *a* represents the height of the triangle, *b* represents the base or one of the sides of the triangle and *h* represents the height of the prism.

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## Examples of triangular prism problems

The following examples are solved by applying the formulas of rectangular prisms seen above.

### EXAMPLE 1

A triangular prism has a base length of 6 m and a height of 5 m. If the height of the prism is 7 m, what is the volume of the prism?

**Solution:** From the question, we have the values $latex b=6$, $latex a=5$ and $latex h=7$. Therefore, we use these values in the volume formula:

$latex V=\frac{1}{2}bah$

$latex V=\frac{1}{2}(6)(5)(7)$

$latex V=105$

The volume of the prism is 105 m³.

### EXAMPLE 2

If a prism has a triangular base with a base of 8 m and a height of 6 m, what is its volume with a height of 10 m?

**Solution:** We substitute the given values in the volume formula:

$latex V=\frac{1}{2}bah$

$latex V=\frac{1}{2}(8)(6)(10)$

$latex V=240$

The volume of the triangular prism is 240 m³.

### EXAMPLE 3

A prism has a base that is an equilateral triangle with sides of length 6 m and a height of 5.2 m. If the height of the prism is 10 m, what is its surface area?

**Solution:** We have the values $latex b=6$, $latex a=5.2$ and $latex h = 10$, so we use these values in the formula for surface area:

$latex A_{s}=ba+3bh$

$latex A_{s}=(6)(5.2)+3(6)(10)$

$latex A_{s}=31.2+180$

$latex A_{s}=211.2$

The surface area is 211.2 m².

### EXAMPLE 4

What is the surface area of a prism that has a height of 9 m and a triangular base with sides of length 12 m and height 10.4 m?

**Solution:** We have the lengths $latex h=9$, $latex b=12$ and $latex a=10.4$. Using these values in the volume formula, we have:

$latex A_{s}=ba+3bh$

$latex A_{s}=(12)(10.4)+3(12)(9)$

$latex A_{s}=124.8+324$

$latex A_{s}=448.8$

The surface area is 448.8 m².

## See also

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