Key Formulas to Find Volume of Rectangular and Triangular Prisms
Calculating the Volume of 3D Figures Prisms Formulas is a core part of solid geometry, essential for understanding shapes in mathematics, science, and engineering. It is frequently tested in school exams, competitive tests like JEE, and even in practical real-life applications such as construction and packaging.
What are Prisms and 3D Figures?
A prism is a solid 3D shape with two identical flat bases and rectangular sides connecting them. Common prisms include rectangular prisms (cuboids), triangular prisms, and others like pentagonal or hexagonal prisms. In general, any solid shape with two parallel congruent faces (bases) joined by parallelogram faces counts as a prism. Prisms are a special category of three-dimensional figures (3D figures), which also include cubes, cylinders, cones, and spheres.
Understanding Volume in Geometry
Volume measures the amount of space occupied inside a 3D figure. It is different from surface area, which is the area covering the shape. Volume is always expressed in cubic units, such as cm³, m³, or liters (L). Accurately finding volume helps in everyday problems like filling tanks or calculating the storage capacity of boxes.
Formulas for Volume of Prisms and Common 3D Figures
The general formula for the volume of any prism is:
Volume = Base Area × Height
| Shape | Base Area Formula | Volume Formula |
|---|---|---|
| Rectangular Prism | l × w | V = l × w × h |
| Triangular Prism | ½ × b × h₁ | V = ½ × b × h₁ × l (or Area of triangle × length/prism height) |
| Cylinder (not strictly a prism, but similar) | π × r² | V = π × r² × h |
| Pentagonal Prism | (5/2) × a × s | V = [(5/2) × a × s] × h |
Here, l = length, w = width, b = base of triangle, h₁ = height of triangle, h = height of prism, a = apothem, s = side length, r = radius, and π ≈ 3.14.
Step-by-Step Worked Examples
Example 1: Volume of a Rectangular Prism (Cuboid)
- Given: length = 10 cm, width = 4 cm, height = 6 cm
- Formula: V = l × w × h
- Calculation: V = 10 × 4 × 6 = 240 cm³
So, the volume is 240 cubic centimeters.
Example 2: Volume of a Triangular Prism
- Triangle base (b) = 5 cm, triangle height (h₁) = 3 cm, prism length = 8 cm
- Base area = ½ × 5 × 3 = 7.5 cm²
- Volume = 7.5 × 8 = 60 cm³
So, the volume is 60 cubic centimeters.
Practice Problems
- Find the volume of a prism with base area 12 cm² and height 10 cm.
- A rectangular box has length 15 cm, width 5 cm, and height 4 cm. What is its volume?
- A triangular prism has a base of 6 cm, triangle height of 4 cm, and prism length 12 cm. Find its volume.
- A cylinder has a radius of 7 cm and height 10 cm. Calculate its volume (use π = 3.14).
- The volume of a prism is 180 m³ and its base area is 20 m². What is its height?
Common Mistakes to Avoid
- Mixing up surface area with volume — surface area is in square units (cm²), volume is in cubic units (cm³).
- Using the wrong base area formula for prisms — always identify the correct base shape.
- For triangular prisms, forgetting to use ½ in base area calculation.
- Not converting all measurements to the same units before calculating.
- Leaving out units in the final answer.
Real-World Applications
Knowing how to calculate volume of prisms is useful in daily life and various professions. For example, architects use these formulas to find building capacities, engineers use them when designing storage containers, and even bakers use them to calculate the batter needed for cake tins shaped as prisms. You might also use these formulas when determining the water a fish tank can hold or in packing and shipping boxes efficiently.
At Vedantu, we break down challenging concepts like Volume of 3D Figures Prisms Formulas into easy parts to help you score higher and solve real problems confidently. To explore more, check related topics like Volume of Cube, Cuboid and Cylinder or Area of a Prism.
By mastering the key formulas for the volume of prisms and understanding how to apply them, you can solve a wide variety of geometry problems, excel in your exams, and use maths confidently in daily life.
FAQs on How to Calculate the Volume of 3D Prisms and Shapes
1. What is the formula for a 3d prism?
The volume of a prism is found by multiplying the area of its base by its height. Volume = Base Area × Height (V = Bh). This applies to all prisms, whether they have a rectangular, triangular, or other polygonal base.
2. How to find the volume of 3 dimensional shapes?
The method depends on the 3D shape. For prisms, use V = Bh (base area times height). For cubes, use V = s³ (side cubed). For cuboids, use V = lwh (length × width × height). Other shapes have different formulas; always identify the shape first.
3. What are the three formulas for volume?
There isn't a universal set of 'three' volume formulas. The formula depends on the 3D shape. Common examples include: V = Bh (for prisms), V = s³ (for cubes), and V = lwh (for cuboids). Many more exist for other shapes like cones, spheres, and pyramids.
4. How to find the volume of a triangular prism?
A triangular prism's volume is calculated using V = Bh, where B is the area of the triangular base. Since the area of a triangle is ½ * base * height, the full formula becomes: V = ½ * base * height * length of the prism (where 'base' and 'height' refer to the triangle's dimensions).
5. What is the volume of a rectangular prism formula?
The volume of a rectangular prism is calculated by multiplying its length (l), width (w), and height (h): V = lwh. This is a specific case of the general prism formula, where the base area is simply length times width.
6. What is the formula of prism volume?
The general formula for the volume of a prism is V = Bh, where 'B' represents the area of the prism's base, and 'h' represents its height. The specific formula for the base area will vary depending on the shape of the base (e.g., triangle, rectangle, etc.).
7. How do you calculate the volume of a prism?
To calculate a prism's volume: 1. Find the area of the prism's base (B). 2. Multiply the base area by the prism's height (h). 3. The result is the volume (V = Bh). Remember to use consistent units throughout the calculation (e.g., cubic centimeters).
8. Volume of a triangular prism formula?
The volume of a triangular prism is given by the formula: V = (1/2) * b * h * l, where 'b' is the base of the triangular face, 'h' is the height of the triangular face, and 'l' is the length of the prism. This formula combines the area of the triangular base with the prism's height.
9. What is volume of 3d figures practice?
Practicing volume calculations for 3D figures involves solving various problems involving different shapes like cubes, cuboids, and prisms. Use the appropriate formula for each shape and focus on understanding the concepts of base area and height. Regular practice with varied examples builds confidence and accuracy.
10. How do I find the volume of irregular 3D shapes?
Finding the volume of irregular 3D shapes is more complex and often requires advanced techniques. Methods include water displacement (measuring the volume of water displaced when the object is submerged) or using integral calculus for more precise measurements. Simple formulas don't apply to these complex shapes.
11. What are the units for volume?
Volume is measured in cubic units. Common units include cubic centimeters (cm³), cubic meters (m³), cubic feet (ft³), and liters (L). The choice of unit depends on the scale of the object being measured.
