What Is the Formula for Volume of Prisms in 3 D Geometry
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 Volume Of 3 D Figures Prisms Explained with Formulas and Examples
1. What is the formula for the volume of a prism?
The formula for the volume of a prism is Volume = Base Area × Height (V = B × h). A prism’s volume depends on the area of its cross-sectional base and the perpendicular height between the two parallel bases.
- B = area of the base
- h = perpendicular height of the prism
2. How do you calculate the volume of a rectangular prism?
The volume of a rectangular prism is calculated using V = l × w × h, where l, w, and h are length, width, and height. Since the base is a rectangle, its area is length × width.
- Step 1: Multiply length × width.
- Step 2: Multiply the result by height.
3. What is the volume formula for a triangular prism?
The volume of a triangular prism is V = (1/2 × b × h) × L, where b and h are the base and height of the triangle, and L is the prism length. First find the area of the triangular base, then multiply by the prism’s length.
- Area of triangle = 1/2 × b × h
- Volume = triangle area × prism length
4. Why is the volume of a prism equal to base area times height?
The volume of a prism equals base area × height because a prism has a constant cross-sectional area throughout its height. This means every horizontal slice has the same area as the base.
- The base area stays the same along the height.
- The height measures how many layers of the base fit vertically.
5. What are the units of volume for prisms?
The units of volume for prisms are always cubic units, such as cm³, m³, or in³. Since volume measures three-dimensional space, all three measurements are multiplied together.
- If dimensions are in cm → volume is in cm³
- If dimensions are in m → volume is in m³
6. How do you find the base area of a prism?
To find the base area of a prism, calculate the area of its base shape using the appropriate area formula. The base can be rectangular, triangular, circular (in cylinders), or any polygon.
- Rectangle: l × w
- Triangle: 1/2 × b × h
- Square: side²
7. What is an example of finding the volume of a prism?
An example of finding the volume of a prism is using V = B × h with actual numbers. Suppose a rectangular prism has length 8 cm, width 3 cm, and height 4 cm.
- Step 1: Base area = 8 × 3 = 24 cm²
- Step 2: Volume = 24 × 4 = 96 cm³
8. What is the difference between the volume of a prism and a pyramid?
The difference is that a prism’s volume is V = B × h, while a pyramid’s volume is V = 1/3 × B × h. A pyramid has one-third the volume of a prism with the same base area and height.
- Prism: uniform cross-section
- Pyramid: tapers to a point
9. Can the base of a prism be any shape?
Yes, the base of a prism can be any polygon as long as the two bases are parallel and congruent. Common examples include triangular, rectangular, pentagonal, and hexagonal prisms.
- The cross-section remains constant.
- Volume formula remains V = B × h.
10. What are common mistakes when finding the volume of prisms?
Common mistakes when finding the volume of prisms include using the wrong base area or forgetting to use perpendicular height. Always apply V = B × h correctly.
- Using slanted height instead of perpendicular height
- Forgetting to find the correct base area first
- Writing square units instead of cubic units
