3D Shapes formulas properties and solved examples
The concept of 3D shapes plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding 3D shapes helps students in geometry, mensuration, engineering, packaging, architecture, and many other fields.
What Is 3D Shapes?
A 3D shape (three-dimensional shape) is a solid object that has three measurements: length, width, and height (or depth). Unlike 2D shapes, which have only length and width, 3D shapes occupy space and have volume. Examples include cubes, cuboids, spheres, cones, cylinders, prisms, and pyramids. You’ll find this concept applied in areas such as surface area, volume calculations, and real-life modeling.
Key Formula for 3D Shapes
Here are some standard formulas for common 3D shapes:
| Shape | Surface Area | Volume |
|---|---|---|
| Cube (side = a) | 6a2 | a3 |
| Cuboid (l, b, h) | 2(lb + bh + lh) | l × b × h |
| Cylinder (r, h) | 2πr(r + h) | πr2h |
| Sphere (r) | 4πr2 | (4/3)πr3 |
| Cone (r, l, h) | πr(l + r) | (1/3)πr2h |
Cross-Disciplinary Usage
3D shapes are not only useful in maths but also play an important role in physics, computer graphics, engineering, architecture, and daily life. Students preparing for exams like JEE, NEET, and Olympiads will often solve problems involving finding the volume or surface area of different 3D objects.
Common 3D Shapes and Their Properties
| Shape | Faces | Edges | Vertices | Example Object |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | Dice |
| Cuboid | 6 | 12 | 8 | Book |
| Cylinder | 3 | 2 (curved) | 0 | Tin can |
| Sphere | 1 (curved) | 0 | 0 | Football |
| Cone | 2 | 1 (curved) | 1 | Ice cream cone |
Step-by-Step Illustration
Let’s solve an example involving a cuboid:
- Given: Length (l) = 10 cm, Breadth (b) = 8 cm, Height (h) = 6 cm
- Volume of cuboid:
V = l × b × h = 10 × 8 × 6 = 480 cm³
- Surface area of cuboid:
SA = 2(lb + bh + lh) = 2(10×8 + 8×6 + 10×6)
= 2(80 + 48 + 60) = 2×188 = 376 cm²
Speed Trick or Vedic Shortcut
To quickly find the volume of a cube with side ‘a’, just remember: Cube the value. For example, if the side is 5 cm, then volume = 5 × 5 × 5 = 125 cm³. Mental math helps a lot with 3D shapes in exams.
Example Trick: If the radius of a sphere is doubled, the volume becomes 8 times bigger. This is a shortcut many students use to answer MCQs fast!
Try These Yourself
- Find the volume and surface area of a cylinder with radius 4 cm and height 10 cm.
- Name a real-life object shaped like a cone.
- Which 3D shape has only one curved face and no edges?
- If the surface area of a cube is 54 cm², what is the length of its edge?
Frequent Errors and Misunderstandings
- Confusing surface area with volume formulas.
- Using 2D shape formulas for 3D problems.
- Forgetting about units (area in cm², volume in cm³).
Relation to Other Concepts
The idea of 3D shapes connects closely with topics like solids and mensuration. Mastery here will help students understand surface area and volume in higher classes. It also helps in symmetry and visualizing solids.
Classroom Tip
To remember the difference between cube and cuboid: Both have 6 faces, 12 edges, and 8 vertices. But all faces of a cube are square, while in a cuboid, faces are rectangles. You can build simple 3D models using building blocks and count the faces, edges, and corners. Vedantu’s teachers use models and nets to help children visualize 3D objects.
We explored 3D shapes—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving maths problems using this concept.
Related Topics and Useful Links
- Solids
- 2D and 3D Figures
- Surface Area and Volume
- Mensuration
- Cube
- Cuboid
- Cylinder
- Sphere
- Cone
- Volume of Cube, Cuboid & Cylinder
FAQs on Understanding 3D Shapes in Geometry
1. What are 3D shapes in Maths?
3D shapes are solid figures that have length, width, and height. Unlike 2D shapes, three-dimensional shapes occupy space and have volume. Examples of common 3D shapes include:
- Cube
- Cuboid
- Sphere
- Cylinder
- Cone
- Pyramid
2. What is the difference between 2D and 3D shapes?
The main difference is that 2D shapes have only length and width, while 3D shapes have length, width, and height. Key differences include:
- 2D shapes are flat (e.g., square, circle).
- 3D shapes are solid and occupy space (e.g., cube, sphere).
- 2D shapes have area only.
- 3D shapes have both surface area and volume.
3. What are faces, edges, and vertices in 3D shapes?
In 3D shapes, faces are flat surfaces, edges are line segments where faces meet, and vertices are corner points where edges meet. For example, in a cube:
- 6 faces
- 12 edges
- 8 vertices
4. What is the formula for the volume of a cube?
The volume of a cube is calculated using the formula V = a³, where a is the length of one side. Since all sides of a cube are equal:
- Multiply side × side × side.
- Example: If side = 4 cm, then V = 4³ = 64 cm³.
5. What is the formula for the volume of a cuboid?
The volume of a cuboid is given by V = l × w × h, where l is length, w is width, and h is height. To calculate:
- Multiply all three dimensions.
- Example: 5 cm × 3 cm × 2 cm = 30 cm³.
6. How do you find the surface area of a cube?
The surface area of a cube is calculated using SA = 6a², where a is the side length. Since a cube has 6 identical square faces:
- Find the area of one face: a².
- Multiply by 6.
- Example: If a = 3 cm, SA = 6 × 3² = 6 × 9 = 54 cm².
7. What is the volume formula for a cylinder?
The volume of a cylinder is given by V = πr²h, where r is the radius of the base and h is the height. To calculate:
- Square the radius.
- Multiply by π (≈ 3.14).
- Multiply by height.
- Example: r = 2 cm, h = 5 cm → V = 3.14 × 4 × 5 = 62.8 cm³.
8. What is the difference between a cube and a cuboid?
A cube has all sides equal, while a cuboid has rectangular faces with possibly different lengths, widths, and heights. Key differences:
- Cube: All edges equal, 6 square faces.
- Cuboid: Opposite faces equal, rectangular faces.
- Cube volume: a³.
- Cuboid volume: l × w × h.
9. What is Euler’s formula for 3D shapes?
Euler’s formula for polyhedra is F + V − E = 2, where F is faces, V is vertices, and E is edges. For example, in a cube:
- F = 6
- V = 8
- E = 12
- 6 + 8 − 12 = 2
10. What are examples of 3D shapes in real life?
Many everyday objects are examples of 3D shapes. Common real-life examples include:
- Dice → Cube
- Brick → Cuboid
- Ball → Sphere
- Can → Cylinder
- Ice cream cone → Cone
