The Volume Of A Triangular Prism Formula states that the volume equals the area of the triangular base multiplied by the length of the prism, expressed as V = (1/2) × b × h × l. This formula is covered in NCERT Mathematics for Class 8 and Class 10, and it also appears in mensuration problems in JEE Main and NEET-based competitive exams. In this article, we cover the complete derivation, a formula sheet for related solids, three solved examples at progressive difficulty levels, CBSE exam tips for 2025-26, and common mistakes to avoid.
Key Volume Of A Triangular Prism Formulas at a Glance
Quick reference for the most important formulas related to a triangular prism.
- Volume of a triangular prism: \( V = \frac{1}{2} \times b \times h \times l \)
- Volume using base area: \( V = A_{\triangle} \times l \)
- Area of triangular base (right triangle): \( A_{\triangle} = \frac{1}{2} \times b \times h \)
- Area of triangular base (equilateral): \( A_{\triangle} = \frac{\sqrt{3}}{4} a^2 \)
- Lateral Surface Area: \( LSA = (a + b + c) \times l \)
- Total Surface Area: \( TSA = LSA + 2 \times A_{\triangle} \)
- Volume using Heron’s formula base: \( V = \sqrt{s(s-a)(s-b)(s-c)} \times l \)
What is the Volume Of A Triangular Prism Formula?
The Volume Of A Triangular Prism Formula gives the total three-dimensional space enclosed within a triangular prism. A triangular prism is a polyhedron that has two identical triangular faces at its ends and three rectangular faces connecting them. The triangular faces are called the bases, and the distance between these two bases is called the length (or height) of the prism.
This concept is introduced in NCERT Mathematics, Class 8, Chapter 11 — Mensuration. It is revisited in Class 10 under Surface Areas and Volumes. The core idea is straightforward: the volume of any prism equals the area of its cross-sectional base multiplied by its length.
For a triangular prism whose base is a triangle with base b and height h, and whose prism length is l, the formula becomes:
\[ V = \frac{1}{2} \times b \times h \times l \]
This is the standard form taught in NCERT. When the triangular base is not a right triangle, students use Heron’s formula to find the base area first, and then multiply by the prism length. Understanding this formula thoroughly builds the foundation for 3D geometry in Class 11 and 12.
Volume Of A Triangular Prism Formula — Expression and Variables
The general form of the volume of a triangular prism uses the area of the triangular base multiplied by the prism’s length:
\[ V = A_{\triangle} \times l \]
For a triangle with base b and perpendicular height h, the area is \( \frac{1}{2}bh \). Substituting this gives the most commonly used form:
\[ V = \frac{1}{2} \times b \times h \times l \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| \( V \) | Volume of the triangular prism | Cubic metres (m³) or cm³ |
| \( A_{\triangle} \) | Area of the triangular base | Square metres (m²) or cm² |
| \( b \) | Base of the triangle | Metres (m) or cm |
| \( h \) | Perpendicular height of the triangle | Metres (m) or cm |
| \( l \) | Length (depth) of the prism | Metres (m) or cm |
| \( a, b, c \) | Sides of the triangular base (scalene/equilateral) | Metres (m) or cm |
| \( s \) | Semi-perimeter of the triangle: \( s = \frac{a+b+c}{2} \) | Metres (m) or cm |
Derivation of the Volume Of A Triangular Prism Formula
The derivation follows directly from the general principle for any prism.
Step 1: Recall that the volume of any prism is given by \( V = \text{Base Area} \times \text{Length} \).
Step 2: For a triangular prism, the base is a triangle. The area of a triangle with base \( b \) and height \( h \) is \( A_{\triangle} = \frac{1}{2} \times b \times h \).
Step 3: Substitute the triangular area into the prism volume formula: \( V = \frac{1}{2} \times b \times h \times l \).
Step 4: This result can be verified by imagining the prism as half of a rectangular cuboid. A rectangular cuboid of dimensions \( b \times h \times l \) has volume \( bhl \). Cutting it diagonally along the length gives two equal triangular prisms, each with volume \( \frac{1}{2}bhl \). This confirms the formula.
Complete Mensuration Formula Sheet for Prisms and 3D Solids
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Volume of Triangular Prism | \( V = \frac{1}{2} b h l \) | b = triangle base, h = triangle height, l = prism length | m³ or cm³ | Class 8, Ch 11; Class 10, Ch 13 |
| Volume of Triangular Prism (Heron’s base) | \( V = \sqrt{s(s-a)(s-b)(s-c)} \times l \) | s = semi-perimeter, a,b,c = sides, l = length | m³ or cm³ | Class 9, Ch 12; Class 10, Ch 13 |
| Volume of Equilateral Triangular Prism | \( V = \frac{\sqrt{3}}{4} a^2 \times l \) | a = side of equilateral triangle, l = length | m³ or cm³ | Class 10, Ch 13 |
| Lateral Surface Area of Triangular Prism | \( LSA = (a + b + c) \times l \) | a,b,c = sides of triangle, l = prism length | m² or cm² | Class 8, Ch 11 |
| Total Surface Area of Triangular Prism | \( TSA = LSA + 2A_{\triangle} \) | LSA = lateral area, \( A_{\triangle} \) = base area | m² or cm² | Class 8, Ch 11 |
| Volume of Rectangular Prism (Cuboid) | \( V = l \times b \times h \) | l = length, b = breadth, h = height | m³ or cm³ | Class 8, Ch 11 |
| Volume of Cylinder | \( V = \pi r^2 h \) | r = radius, h = height | m³ or cm³ | Class 9, Ch 13; Class 10, Ch 13 |
| Volume of Cone | \( V = \frac{1}{3} \pi r^2 h \) | r = radius, h = height | m³ or cm³ | Class 9, Ch 13; Class 10, Ch 13 |
| Volume of Sphere | \( V = \frac{4}{3} \pi r^3 \) | r = radius | m³ or cm³ | Class 9, Ch 13; Class 10, Ch 13 |
| Volume of Pyramid (triangular base) | \( V = \frac{1}{3} A_{\triangle} \times H \) | \( A_{\triangle} \) = base area, H = perpendicular height | m³ or cm³ | Class 10, Ch 13 |
Volume Of A Triangular Prism Formula — Solved Examples
Example 1 (Class 8-9 Level — Direct Application)
Problem: A triangular prism has a right-angled triangular base with base 6 cm and height 4 cm. The length of the prism is 10 cm. Find its volume.
Given: b = 6 cm, h = 4 cm, l = 10 cm
Step 1: Write the formula: \( V = \frac{1}{2} \times b \times h \times l \)
Step 2: Substitute the values: \( V = \frac{1}{2} \times 6 \times 4 \times 10 \)
Step 3: Calculate step by step: \( V = \frac{1}{2} \times 240 = 120 \) cm³
Answer
Volume of the triangular prism = 120 cm³
Example 2 (Class 10-11 Level — Scalene Triangle Base)
Problem: A swimming pool has a cross-section in the shape of a triangle with sides 5 m, 12 m, and 13 m. The pool is 25 m long. Find the volume of water it can hold.
Given: a = 5 m, b = 12 m, c = 13 m, l = 25 m
Step 1: Check if the triangle is a right triangle. Since \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \), this is a right-angled triangle with legs 5 m and 12 m.
Step 2: Find the area of the triangular base: \( A_{\triangle} = \frac{1}{2} \times 5 \times 12 = 30 \) m²
Step 3: Apply the volume formula: \( V = A_{\triangle} \times l = 30 \times 25 \)
Step 4: Calculate: \( V = 750 \) m³
Answer
Volume of water the pool can hold = 750 m³
Example 3 (JEE/NEET Level — Equilateral Triangle Base with Surface Area)
Problem: A triangular prism has an equilateral triangular base with side 8 cm. The total surface area of the prism is 288 + 32√3 cm². Find the volume of the prism.
Given: a = 8 cm (equilateral triangle), TSA = \( 288 + 32\sqrt{3} \) cm²
Step 1: Find the area of the equilateral triangular base: \( A_{\triangle} = \frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3} \) cm²
Step 2: Write the TSA formula: \( TSA = (a + b + c) \times l + 2A_{\triangle} \). For an equilateral triangle, \( a + b + c = 3 \times 8 = 24 \) cm.
Step 3: Substitute into TSA: \( 288 + 32\sqrt{3} = 24l + 2 \times 16\sqrt{3} \)
Step 4: Simplify: \( 288 + 32\sqrt{3} = 24l + 32\sqrt{3} \)
Step 5: Solve for l: \( 24l = 288 \), so \( l = 12 \) cm
Step 6: Calculate the volume: \( V = A_{\triangle} \times l = 16\sqrt{3} \times 12 = 192\sqrt{3} \) cm³
Step 7: Approximate if needed: \( V \approx 192 \times 1.732 \approx 332.5 \) cm³
Answer
Volume of the triangular prism = \( 192\sqrt{3} \) cm³ ≈ 332.5 cm³
CBSE Exam Tips 2025-26
- Identify the base correctly: Always identify which face is the triangular base before applying the formula. CBSE questions sometimes orient the prism differently in diagrams.
- Check for right-angled triangles first: Before using Heron’s formula, check if the triangle satisfies the Pythagorean theorem. A right-angled base saves significant calculation time in exams.
- Use Heron’s formula for scalene bases: When all three sides of the base are given but no height is provided, use \( s = \frac{a+b+c}{2} \) and then \( A = \sqrt{s(s-a)(s-b)(s-c)} \).
- Keep units consistent: We recommend converting all measurements to the same unit before substituting. Mixing cm and m is the most common error in CBSE exams.
- Write the formula first: In CBSE board exams 2025-26, always write the formula before substituting values. Examiners award one mark for the correct formula, even if the final calculation has a minor error.
- Practise both TSA and Volume together: CBSE often combines surface area and volume in a single 4-5 mark question. Our experts suggest revising both formulas together for maximum efficiency.
Common Mistakes to Avoid with the Volume Of A Triangular Prism Formula
- Mistake 1 — Forgetting the ½ factor: Many students write \( V = b \times h \times l \) instead of \( V = \frac{1}{2} \times b \times h \times l \). The factor of ½ comes from the area of the triangle and must always be included when using the base and height of the triangle directly.
- Mistake 2 — Confusing prism height with triangle height: A triangular prism has two different heights — the height of the triangular base (\( h \)) and the length of the prism (\( l \)). These are completely different measurements. Substituting one for the other doubles or halves the answer incorrectly.
- Mistake 3 — Applying the formula to a triangular pyramid: A triangular prism and a triangular pyramid look similar in 2D diagrams. The prism has two triangular faces and three rectangular faces. The pyramid has one triangular base and three triangular faces meeting at a point. The pyramid formula has an extra \( \frac{1}{3} \) factor.
- Mistake 4 — Incorrect unit conversion: If the base is given in cm but the prism length is in metres, students must convert before multiplying. Failing to do so gives an answer in mixed units, which is incorrect.
- Mistake 5 — Using slant height instead of perpendicular height: When finding the area of the triangular base, always use the perpendicular height of the triangle, not the slant side. Using the slant side overestimates the base area and gives a wrong volume.
JEE/NEET Application of Volume Of A Triangular Prism Formula
In our experience, JEE aspirants encounter the volume of a triangular prism in two main contexts: direct mensuration problems and application-based problems involving density, pressure, or optics.
Application Pattern 1 — Density and Mass Problems (JEE Main)
JEE Main frequently tests the formula by combining it with density. A typical problem gives a prism made of a material with known density \( \rho \) (kg/m³) and asks for the mass. The approach is:
\[ \text{Mass} = \rho \times V = \rho \times \frac{1}{2} b h l \]
Students must correctly identify the triangular cross-section, compute its area, multiply by length, and then multiply by density. Errors in the ½ factor directly lead to incorrect mass values.
Application Pattern 2 — Optical Prism in Physics (JEE Advanced / NEET)
In optics, a glass prism used to study refraction and dispersion is a triangular prism. While JEE Advanced focuses on the angle of prism and refractive index rather than volume, some problems do ask for the mass of glass used to make a prism. Here, the volume of a triangular prism formula is essential. NEET occasionally includes such problems under properties of light.
Application Pattern 3 — Combined Solids (JEE Main / CBSE Class 10)
A common problem type presents a solid made by attaching a triangular prism to a rectangular cuboid or a cylinder. Students must find the total volume by adding the individual volumes. Recognising the triangular prism component and applying \( V = \frac{1}{2}bhl \) correctly is the key skill tested. In our experience, JEE aspirants who practise Heron’s formula alongside this formula solve such combined-solid problems significantly faster during the exam.
FAQs on Volume Of A Triangular Prism Formula
For more related formulas, explore our comprehensive guides on the Algebra Formulas hub, the 30-60-90 Triangle Formula for special right triangles used in equilateral prism problems, and the Degree and Radian Measure Formula for angle-based geometry. You can also visit the official NCERT website to download the Class 8 and Class 10 Mathematics textbooks for additional practice problems on mensuration and surface areas and volumes.