Triangular Prism

• Introduction
• What is a Triangular Prism?
• Properties of Triangular Prism
• Types of Triangular Prism
• Triangular Prism Net
• Surface Area of a Triangular Prism
• Volume of a Triangular Prism
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

A prism is a three-dimensional polyhedron. A prism typically consists of two identical bases which are `n`-sided polygons, and `n` other faces, necessarily all parallelograms, joining the corresponding sides of the two bases. The name of a particular prism depends on the two bases of the prism which can be triangles, rectangles, or any `n`-sided polygon. For example, a prism with triangular bases is called a triangular prism and a prism with a square base is called a square prism, and so on. In this topic, we are going to learn about various properties of a triangular prism and more.

What is a Triangular Prism?

A triangular prism is a `3D` shape that has two identical triangular bases joined together by three rectangular sides. These rectangular faces are called the lateral faces of a triangular prism. The triangular bases are also referred to as the top and bottom faces of the prism. The ends of the prism are parallel, meaning they're the same distance apart all the way along. And they're also congruent, which means they're the same shape and size. If you count up all the faces, you get `5`. For the corners where the faces meet, you have `6` of those; where the edges come together, there are `9`.

In simpler terms, a triangular prism has five faces, nine edges, and six vertices. Both triangular faces are the same size, and all three rectangular faces are identical.

When describing its size, we use terms like the length of the prism `(l)`, the height of the triangular base `(h)`, and the length of the bottom edge of the triangular base `(b)`.

Properties of Triangular Prism

The properties of a triangular prism help us recognize it easily. Let's break down the properties into some key points:

• Number of Faces: A triangular prism has five faces: two are triangular bases, and the other three are rectangular sides.
• Number of Edges: It has nine edges, which are where the faces meet. The prism's edges include the `6` edges formed by the two triangular bases `(3 + 3)`, along with an additional `3` edges created by the sides connecting the bases.
• Number of Vertices: A triangular prism has six vertices, the points where the edges meet. The vertices of the prism coincide with the vertices of its two triangular bases, connected by lines that form rectangles along the prism’s sides.
• Polyhedral Nature: It's a type of polyhedron characterized by two triangular bases and three rectangular sides. 
• Base Shape: The base of a triangular prism is a triangle.
• Side Shape: The sides are rectangles, maintaining a consistent structure along the length of the prism.
• Equilateral Triangular Bases: Both triangular bases are equilateral triangles, meaning all their sides are of equal length.
• Cross-Section Shape: Any cross-section of a triangular prism cut, along its base, forms a triangle.
• Congruent Bases: The two triangular bases are identical to each other, indicating their congruence.

Types of Triangular Prism

When we talk about the types of triangular prism, we categorize them based on two factors: uniformity and alignment.

Based on uniformity, triangular prisms fall into two categories:

Regular Triangular Prism:

This is a prism where both triangular bases are regular triangles. In simpler terms, all the sides of these triangles are equal, and the angles between them measure `60` degrees. The sides of the prism are rectangular.

Irregular Triangular Prism:

Here, at least one of the triangular bases isn't a regular triangle. It means the sides of the base triangle can have different lengths, and the angles between them might not be fixed at 60 degrees. The lateral faces are still rectangles.

Based on alignment, triangular prisms can be divided into two types:

Right Triangular Prism: 

In this type, the angle between the edges of the triangular bases and the edges of the rectangular faces is exactly `90` degrees. So, the bases meet the rectangular faces at right angles. The other properties of the prism remain the same.

Oblique Triangular Prism: 

Here, the lateral faces are not perpendicular to the bases. Instead of rectangles, they form parallelograms. This means the angles between the lateral faces and the bases might not be `90` degrees. This allows for more flexibility in the prism's geometric configuration.

Triangular Prism Net

The net of a triangular prism is like a blueprint that unfolds the surface of the prism. When you open, flatten, and lay out the prism's surface, you can see all its faces. This resulting pattern is a two-dimensional figure known as the net. Folding this net allows you to recreate the original triangular prism. The net clearly illustrates that the prism has triangular bases and rectangular lateral faces. In simpler terms, it serves as a visual guide showing how the prism can be assembled from a flat, folded shape.

Surface Area of a Triangular Prism

The surface area of a triangular prism is the total area covered by its surface. It's made up of the areas of all the `5` faces of the prism. To calculate it, we use a formula that considers both the lateral surface area and the area of the bases.

Lateral Surface Area (LSA)

For the lateral surface area (LSA), we add up the areas of all the side faces, excluding the top and bottom faces. This is given by the formula: \( \text{LSA} = (s_1 + s_2 + b) \times l \), where \( s_1 \) and \( s_2 \) are the lengths of the edges of the base triangle, and \( b \) is the measure of the base of the triangle, while \( l \) is the length of the prism.

Total Surface Area (TSA)

The total surface area (TSA) includes the lateral surface area and twice the area of one of the triangular bases. 

`"Area of one triangular base" = 1/2* b* h` where `b` and `h` are the base and height of the triangular base.

`"Area of 2 triangular bases" = 2 * (1/2 * b* h) = b * h`

Accordingly, the formula is \( \text{TSA} = (b \times h) + (s_1 + s_2 + b) \times l \), where \( s_1, s_2, \) and \( b \) are the edges of the triangular base, \( h \) is the height of the base triangle and \( l \) is the length of the prism.

Alternatively, we can simplify the formula for a right triangular prism to \( \text{TSA} = (s_1 + s_2 + h) \times L + b \times h \), where \( b \) represents the bottom edge of the base triangle.

Volume of a Triangular Prism

The volume of a triangular prism is the amount of space it takes up in three-dimensional space. To find the volume, we multiply the area of the triangular base by the length of the prism. Since the base is a triangle, its area is found using the formula for the area of a triangle. 

Therefore, the formula for the volume of a triangular prism is given by: \( \text{Volume} = \frac{1}{2} \times \text{base length} \times \text{height} \times \text{length} \).

, the base edge represents the length of one of the edges forming the base triangle, the height of the triangle is the perpendicular distance from the base to the opposite vertex, and the length of the prism indicates the overall length along its axis. By plugging these values into the formula, we can calculate the volume of the triangular prism.

Solved Examples

Example `1`. Find the volume of a triangular prism with a base length of \(8\) units, a height of \(5\) units, and a length of \(12\) units.

Solution:

To find the volume of the triangular prism, we use the formula \(V = \frac{1}{2} \times b \times h \times l\), where \(b\) is the base length, \(h\) is the height, and \(l\) is the length:

\(V = \frac{1}{2} \times 8 \times 5 \times 12\)

\(V = \frac{1}{2} \times 8 \times 5 \times 12\)

\(V = 240 \text{ cubic units}\)

Therefore, the volume of the triangular prism is \(240\) cubic units.

Example `2`. Calculate the surface area of a triangular prism with a base area of \(15\) square units, a length of \(8\) units, and a perimeter of the base as \(20\) units.

Solution: 

We compute the surface area of the prism using the formula:

\(\text{Surface area} = (\text{Perimeter of the base} \times \text{Length}) + (2 \times \text{Base Area})\)

`"Surface area" = (20 \times 8) + (2 \times 15)`

`"Surface area" = 160 + 30`

`"Surface area" = 190` square units

Thus, the surface area of the triangular prism is \(190\) square units.

Example `3`. Find the volume of a triangular prism with a base area of \(35\) \(cm^2\) and a length of \(12\) cm.

Solution: 

Using the formula `"Volume" = "base area" × h × l`, where \(b\) is the base, \(h\) is the height, and \(l\) is the length:

\( \text{Volume} = 35 \times 12\)

\( \text{Volume} = 420 \text{ cubic cm}\)

Example `4`. Find the base length of a triangular prism with the volume of the prism is \(72\) units, with a length of \(8\) units and a base height of \(6\) units.

Solution: 

Using the volume formula \(V = \frac{1}{2} \times b \times h \times l\), we solve for the base length \(b\):

\(72 = \frac{1}{2} \times b \times 6 \times 8\)

\(144 = b \times 48\)

\(b = \frac{144}{48} = 3\)

Thus, the base length of the triangular prism is \(3\) units.

Example `5`. Compute the total surface area of a regular triangular prism whose length is \(9\) cm. The sides of the equilateral triangular base measure \(12\) cm and its height is \(7\) cm.

Solution: 

Using the formula `"Surface area" = (sh + (3s)l)` where \(h\) is the height of the base, \(s\) is the sides of the base, and \(l\) is the length of the prism:

`"Surface area" = (12 × 7 + (3 *12) × 9)`

`"Surface area" = (84 + 324)`

`"Surface area" = 408` square cm

Thus, the total surface area of the prism is \(408\) square centimeters.

Practice Problems

Q`1`. What is the volume of a triangular prism with a base length of \(7\) units, a height of \(4\) units, and a length of \(10\) units?

1. \(140\) cubic units
2. \(150\) cubic units
3. \(160\) cubic units
4. \(170\) cubic units

Answer: a

Q`2`. Calculate the surface area of a triangular prism if its base area is \(24\) square units, length is \(8\) units, and the perimeter of the base is \(18\) units.

1. \(230\) square units
2. \(192\) square units
3. \(245\) square units
4. \(162\) square units

Answer: b

Q`3`. Find the length of the base of a triangular prism if its volume is \(96\) cubic units, the height of the triangular base is \(6\) units, and the length of the prism is \(8\) units.

1. \(8\) units
2. \(10\) units
3. \(12\) units
4. \(4\) units

Answer: d

Q`4`. Determine the lateral surface area of a regular triangular prism with a base length of \(9\) units, a height of \(5\) units, and a length of \(12\) units.

1. \(284\) square units
2. \(324\) square units
3. \(264\) square units
4. \(270\) square units

Answer: b

Q`5`. Calculate the total surface area of a triangular prism with a base length of \(6\) units, a height of \(8\) units, and a length of \(15\) units.

1. \(312\) square units
2. \(322\) square units
3. \(332\) square units
4. \(342\) square units

Answer: d

Q`6`. A triangular prism has a volume of \(126\) cubic meters. The base and height of the triangular prism are \(14\) meters and \(6\) meters respectively. Find the length of the prism.

1. \(3\) units
2. \(4\) units
3. \(5\) units
4. \(6\) units

Answer: a

Frequently Asked Questions

Q`1`. What is a triangular prism?

Answer: A triangular prism is a three-dimensional geometric shape characterized by two congruent triangular bases and three rectangular or parallelogram faces. It has six vertices and nine edges.

Q`2`. How do you find the volume of a triangular prism?

Answer: The volume of a triangular prism can be calculated using the formula: `"Volume" = 1/2 × "base length" × "height" × "prism length"`, where the base length refers to the length of one side of the triangular base, height of the triangular base, and length represents the overall length of the prism.

Q`3`. What are the properties of a triangular prism?

Answer: The key properties of a triangular prism include having two congruent triangular bases, three rectangular (or parallelogram) faces, six vertices, and nine edges. It can also be classified based on uniformity (regular or irregular) and alignment (right or oblique).

Q`4`. How do you calculate the total surface area of a triangular prism?

Answer: To find the total surface area of a triangular prism, you need to calculate the areas of all its faces and then sum them up. The formula for the total surface area of a triangular prism is `"Surface Area" = ("Perimeter of the base" × "Length") + (2 × "Base Area")`, where the base area is the area of one of the triangular bases.

Q`5`. What are some real-life examples of triangular prisms?

Answer: Triangular prisms are commonly found in various everyday objects and structures. Examples include roofs of houses, camping tents, certain types of packaging, pyramids with triangular bases, certain architectural elements, and the design of some musical instruments like certain types of bells.