The right triangular prism is one of the most important subjects studied in space engineering. It is a three-dimensional shape, which often reminds us of the shape of a Toblerone chocolate bar. This article will focus on the structure of the right triangular prism, its properties, and the ways in which we can calculate its area and volume.

Another important data that can be obtained when working with right triangular prisms is the volume. Since it is a three-dimensional geometric figure, it is possible to quantify the space it occupies. The volume of a right triangular prism can be found by multiplying the area of its base by the length of its height.

In any right triangular prism, it is possible to calculate the surface area. In general, it can be said that to calculate the surface area of a right triangular prism, we must calculate the area of each of its bases and the area of each of its three lateral faces~~,~~ and then add these measurements together. The total amount obtained is, in fact, the surface area of the prism.

The area of the lateral surface of a right triangular prism is calculated by adding the areas of its lateral faces, that is, without the area of the bases.

We have a right triangular prism, in which the two bases are right triangles.

We are given the following data:

The length of the legs of the right triangle, is $6$ and $8$ cm.

The length of the hypotenuse of the right triangle is $10$ cm.

The height of the prism is $12$ cm.

Use these data and the attached drawing to calculate the following:

1- The volume of the prism

2- The area of the lateral surface of the prism.

3- Calculate the surface area of the prism.

**Solution:**

1- As we have already learned, to calculate the volume of the prism, multiply the height of the prism by the area of one of its bases.

Look carefully at the drawing and the data, and you will see that each of the bases of the prism is actually a right triangle.

Calculate the area of the right triangle by multiplying the legs and dividing the product by $2$.

**Which gives us:**

$S=\frac{8X6}{2}$

$S=\frac{48}{2}$

$S=24$

That is, the area of the base of the right triangular prism is $24$ cm².

Now we have to calculate the volume by multiplying the area of the base that we already obtained, by the height of the prism.

**We will do the following operation:**

$V=24X12=288$

That is, the volume of the right triangular prism is $288$ cm³.

2- To calculate the area of the lateral surface of the right triangular prism, we actually need to add the areas of the three rectangles that are part of the prism.

Rectangle number .1: sides $12$ cm and $6$ cm

Rectangle number 2: sides $12$ cm and $8$ cm

Rectangle number 3: sides $12$ cm and $10$ cm

Now, we will calculate the surface area of each of these three rectangles:

$S1=12X6=72$

$S2=12X8=96$

$S3=12X10=120$

We add up all the areas and we will get:

$S=S1+S2+S3=72+96+120=288$

Therefore the lateral surface area of the right triangular prism is $228$ cm².

3- To calculate the surface area of our prism, we have to add the area of the lateral surface (which we calculated previously), plus the area of the two bases.

Previously we have calculated the area of one of the bases (right triangle), and now we have to multiply this result by two, since the two triangular bases are identical to each other. Therefore, if the area of the base of the right triangular prism is $24$ cm², then the sum of the area of the bases is equal to $48$ cm² (24X2).

Now, we have to add the sum of the bases, to the area of the lateral surface (which we have already calculated). Thus we will obtain:

$S=48+288=336$

The surface area of the right triangular prism is $336$ cm².

**Answer:**

- The volume of the right triangular prism is $288$ cm³.
- The area of the lateral surface is $288$ cm².
- The surface area of the right triangular prism is $336$ cm².

**What is the definition of a triangular prism?**

A triangular prism or triangular base prism is a prism that has two faces (bases) in the shape of triangles.

**What is a triangular base prism?**

A triangular base prism as its name indicates is a prism that has two equal faces in the form of triangles.

**How many lateral faces does a triangular prism have?**

A triangular prism has three lateral faces.

**What geometric figures form the triangular prism?**

The triangular prism is formed by two equal triangles (bases) and by three rectangles (lateral faces).

**What are the properties of a prism?**

It has two equal and parallel sides called bases, and the other sides called lateral faces are parallelograms.

**What are the different types of prisms that exist?**

Prisms can be classified depending on the number of sides of the bases, thus we can find triangular prisms, quadrangular prisms, pentagonal prisms, etc.

It is also possible to classify prisms by the shape of the base, i.e. if the base is a regular polygon we will have a regular prism, while if the base is an irregular polygon we will have an irregular prism.

For example if the base is an equilateral triangle we have a regular and triangular prism.

**How many edges does a prism with a triangular base have?**

A triangular prism has $9$ edges.

**How to get the edge of a triangular prism?**

To calculate the number of edges of a prism in general, it is enough to multiply by three the number of edges of the base.

For the particular case of a triangular prism, the number of sides of the base is $3$, therefore, multiplying by $3$, we obtain that it has $9$ edges.

**How to calculate the area of a prism with a triangular base?**

The area of the triangular base is calculated, using the formula "base times height over two". The result obtained is multiplied by the height of the prism.