How to Find the Volume and Surface Area of a Triangular Prism?
The concept of triangular prism plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding triangular prisms helps students handle geometry problems with confidence, especially when working with solid shapes, volume, and surface area calculations. This topic is regularly seen in school exams and competitions, and also appears in Science and Engineering contexts.
What Is a Triangular Prism?
A triangular prism is a 3D solid shape that has two identical triangular faces (called bases) and three rectangular faces (called lateral faces). All together, it has 5 faces, 9 edges, and 6 vertices. You’ll find this concept applied in areas such as geometry solids, nets of solid shapes, and real-life applications like rooftops, tents, and chocolate bars.
Key Formula for Triangular Prism
Here’s the standard formula:
Volume of Triangular Prism: \( V = \dfrac{1}{2} \times b \times h \times l \), where
b = base of triangle
h = height of triangle (not prism!)
l = length of prism (distance between the triangular bases)
Surface Area of Triangular Prism:
\( SA = (a + b + c) \times l + b \times h \), where
l = length of prism
b × h = area of one base triangle
Triangular Prism Properties (Faces, Edges, Vertices)
| Triangular Prism | Number |
|---|---|
| Faces | 5 |
| Edges | 9 |
| Vertices | 6 |
Net of a Triangular Prism
The net of a triangular prism is a flat, 2D shape that can be folded to make the prism. The net shows two triangles and three rectangles. Recognizing the net helps in learning how the faces fit together—a key skill for MCQs and visual reasoning. For more solids, see our Nets of Solid Shapes page.
Step-by-Step Illustration: Volume of a Triangular Prism
- Write down given dimensions:
Base of triangle, b = 5 cm
Height of triangle, h = 3 cm
Length (Height of prism), l = 8 cm - Find the area of the triangular base:
Area = (1/2) × 5 × 3 = 7.5 cm² - Multiply area by length:
Volume = 7.5 × 8 = 60 cm³ - Final Answer: 60 cm³
Cross-Disciplinary Usage
Triangular prisms are not only useful in Maths but also play an important role in Physics, Computer Science, and logical reasoning. For example, knowledge of prisms helps explain light refraction in optics, analyze bridge designs in Engineering, and solve 3D visualization problems in coding competitions. Students preparing for JEE or NEET often find questions on 3D solids like triangular prisms.
Solved Examples: Triangular Prism
Example 1: Find the Volume
Given:
Base, b = 4 cm, Height, h = 3 cm, Length of prism, l = 6 cm
Steps:
1. Area of base = (1/2) × 4 × 3 = 6 cm²
2. Volume = 6 cm² × 6 cm = 36 cm³
Final Answer: 36 cm³
Example 2: Surface Area Calculation
Given: Sides of base triangle = 5 cm, 6 cm, 7 cm; Length of prism = 10 cm; Height of base = 4 cm
1. Perimeter = 5 + 6 + 7 = 18 cm
2. Base area = (1/2) × 5 × 4 = 10 cm²
3. Surface Area = (Perimeter × Length) + (2 × Base area)
= (18 × 10) + (2 × 10) = 180 + 20 = 200 cm²
Speed Trick or Vedic Shortcut
To quickly compare the number of faces or edges of prisms, just remember: a prism always has 2 × (number of base sides) for its lateral edges—so a triangular prism has 3 × 2 = 6 base edges, plus 3 more connecting the triangles, for a total of 9 edges. Tricks like these save time in competitive exams. Vedantu’s live classes teach many more such tricks to make geometry faster and easier!
Prism vs Pyramid: Key Differences
| Feature | Triangular Prism | Triangular Pyramid |
|---|---|---|
| Bases | 2 triangles | 1 triangle |
| Other Faces | 3 rectangles | 3 triangles |
| Total Faces | 5 | 4 |
| Edges | 9 | 6 |
| Vertices | 6 | 4 |
Real-life Applications of Triangular Prism
Some common examples of triangular prisms in real life are:
• Triangular rooftops
• Chocolate bars like Toblerone
• Bridge structures
• Light-refracting glass prisms in Physics labs
Try These Yourself
- Draw the net of a triangular prism and mark all faces.
- Calculate the surface area if the base sides are 3 cm, 4 cm, and 5 cm, and the length is 7 cm.
- Find the number of vertices, faces, and edges of a triangular prism.
- List three real-life objects shaped like a triangular prism.
Frequent Errors and Misunderstandings
- Mixing up the height of the prism (length) with the height of the base triangle.
- Forgetting to use (1/2) in the base area calculation.
- Confusing triangular prisms with pyramids or cubes.
- Not counting the number of faces correctly—always use a net to double-check!
Relation to Other Concepts
The idea of a triangular prism connects closely with topics like cuboid (rectangular prism), solid geometry, and area and perimeter. Mastering this concept helps students move on to tougher solids, surface area problems, and advanced shape comparisons.
Classroom Tip
A quick way to remember a triangular prism: it’s like a Toblerone chocolate bar! Two same triangles at the front and back, rectangles all around. Use colors to shade the faces differently when revising, or make a folded paper net in class. Vedantu’s expert teachers use 3D models to help students visualize and remember solid shapes.
We explored triangular prisms—from its definition, properties, formula, solved examples, real-life uses, and how to avoid common errors. Keep practicing nets and volume/surface area problems to score full marks in this topic. For more concepts like this, practice with Vedantu or check related links below!
Explore related topics:
Cuboid and Cube |
Nets of Solid Shapes |
Volume of Cuboid |
Solid Geometry
FAQs on Triangular Prism: Definition, Formula & Solved Problems
1. What is a triangular prism in maths?
A triangular prism is a three-dimensional geometric shape with two identical triangular bases and three rectangular lateral faces. It's a type of polyhedron. Key features include:
- Two congruent and parallel triangular bases
- Three rectangular lateral faces connecting the bases
- 5 faces in total
- 9 edges
- 6 vertices
2. How do you calculate the volume of a triangular prism?
The volume of a triangular prism is calculated by multiplying the area of its triangular base by its height (length between the bases). The formula is: Volume = (1/2 × base × height of triangle) × length of prism, often written as V = (1/2bh)l where 'b' is the base of the triangle, 'h' is the height of the triangle, and 'l' is the length of the prism.
3. How many faces, edges, and vertices does a triangular prism have?
A triangular prism has 5 faces (two triangular bases and three rectangular lateral faces), 9 edges, and 6 vertices.
4. What is the difference between a triangular prism and a pyramid?
A triangular prism has two parallel triangular bases connected by rectangular faces. A triangular pyramid (or tetrahedron) has one triangular base and three triangular faces meeting at a single apex. Key differences lie in the number of faces and the overall shape.
5. What does a triangular prism net look like?
The net of a triangular prism is a two-dimensional representation showing how the prism's faces unfold. It consists of two congruent triangles (the bases) and three rectangles (the lateral faces) arranged to form the prism when folded.
6. How is the surface area of a triangular prism calculated?
The surface area is the sum of the areas of all its faces. The formula is: Surface Area = (Perimeter of base × length) + (2 × Area of base). You'll need to calculate the area of one triangular base and add that twice to the area of all three rectangular lateral faces.
7. What are some real-life examples of triangular prisms?
Triangular prisms are found in various everyday objects. Some examples include: tent structures, certain types of roofs, some candy bar packaging (like Toblerone), and parts of certain architectural designs.
8. What is a right triangular prism?
A right triangular prism is a specific type where the lateral faces are perpendicular to the bases. This means the triangular bases are directly above each other, and the lateral faces are rectangles.
9. How do you find the base area of a triangular prism?
The base area is simply the area of one of its triangular bases. Use the standard formula for a triangle's area: Area = (1/2) × base × height, where the base and height are dimensions of the triangular base of the prism.
10. Can you explain the difference between oblique and right triangular prisms?
In a right triangular prism, the lateral edges are perpendicular to the bases. In an oblique triangular prism, the lateral edges are not perpendicular to the bases; they are slanted, resulting in parallelogram-shaped lateral faces instead of rectangles.
11. What are some common mistakes students make when calculating triangular prism volume and surface area?
Common mistakes include: forgetting to multiply by 1/2 when calculating the area of the triangular base for volume, incorrectly identifying the base or height in the area calculations, and using the wrong formula for either volume or surface area altogether. Double-checking your measurements and formulas can help.
12. How can I visualize the net of a triangular prism more easily?
Imagine carefully cutting open a real or model triangular prism along its edges. Laying it flat will create the net. Practice drawing nets of triangular prisms by starting with the two triangular bases and then adding the rectangular lateral faces to connect them. Using colorful markers to represent different faces can also help with visualization.
