What are the Properties and Formulas of Three Dimensional Shapes?
The concept of three dimensional shapes plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From understanding solids like cubes, spheres, and cylinders to solving problems about surface area and volume, three dimensional shapes appear frequently in school maths and in practical fields such as engineering, architecture, and design.
What Is Three Dimensional Shape?
A three dimensional shape (often called a 3D shape) is defined as a solid object that has three measurable dimensions: length, width, and height. These geometric solids occupy space, unlike two dimensional (2D) shapes which only have length and width. Examples of three dimensional shapes include cubes, cuboids, spheres, cylinders, cones, prisms, and pyramids. You’ll find this concept applied in topics like geometry, mensuration, and visualising solid shapes.
Key Properties of Three Dimensional Shapes
Each 3D shape has certain properties that help us to identify and work with them. The most important are:
- Faces: Flat or curved surfaces on the shape
- Edges: Lines where two faces meet
- Vertices: Points where edges meet (corners)
Some shapes like spheres or cylinders also have only curved surfaces, while others like cubes have only flat faces. Recognising these features is the first step in distinguishing different 3D shapes.
List of Common Three Dimensional Shapes
| Name | Faces | Edges | Vertices | Example in Daily Life |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | Dice, Ice Cube |
| Cuboid | 6 | 12 | 8 | Book, Box |
| Sphere | 1 (curved) | 0 | 0 | Football, Globe |
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 | Can, Pipe |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 | Ice Cream Cone, Party Hat |
| Pyramid | 5 (Square base) | 8 | 5 | Egyptian Pyramid |
| Prism | Depends | Depends | Depends | Tent, Toblerone Chocolate |
Standard Formulas for Three Dimensional Shapes
| Shape | Surface Area | Volume |
|---|---|---|
| Cube | \(6a^2\) | \(a^3\) |
| Cuboid | \(2(lw + lh + wh)\) | \(l \times w \times h\) |
| Sphere | \(4\pi r^2\) | \(\frac{4}{3}\pi r^3\) |
| Cylinder | \(2\pi r(h + r)\) | \(\pi r^2 h\) |
| Cone | \(\pi r(r + l)\) | \(\frac{1}{3}\pi r^2 h\) |
| Pyramid | Base Area + (1/2 × Perimeter × Slant Height) | \(\frac{1}{3}\) × Base Area × Height |
| Prism | Depends on type | Base Area × Height |
Where a = side of cube, l = length, w = width, h = height, r = radius, l (in cone) = slant height. Always use the right formula for the given 3D solid!
Difference Between 2D and 3D Shapes
| 2D Shapes | 3D Shapes |
|---|---|
| Have only length and width (e.g., square, triangle) |
Have length, width, and height (e.g., cube, sphere) |
| Flat and do not occupy space | Solid, occupy space (have volume) |
| Measured in square units | Measured in cubic units |
Step-by-Step Illustration
Let’s solve a problem about the surface area of a cuboid:
1. Write the formula: Total Surface Area = \(2(lw + lh + wh)\)2. Insert the given values: Let \(l = 4\), \(w = 3\), \(h = 6\)
3. Calculate each product:
4 × 6 = 24
3 × 6 = 18
4. Add: 12 + 24 + 18 = 54
5. Multiply by 2: 2 × 54 = 108
6. Final Answer: The total surface area is 108 square units.
Speed Trick or Vedic Shortcut
When asked to quickly estimate volume of a cube if only one edge is given, simply cube the side length: For example, if side = 5, then volume = 5 × 5 × 5 = 125. You don’t have to write the multiplication three times in the exam—practice makes this mental calculation super fast! Tricks like these are shared by Vedantu experts in live classes so you get faster at solving maths questions.
Try These Yourself
- Name any four three dimensional shapes you see in your house.
- Find the volume of a cylinder with radius 3 units, height 10 units.
- List out the number of faces, edges, and vertices for a cone.
- What is the difference between a prism and a pyramid?
- Draw and label the net of a cube.
Frequent Errors and Misunderstandings
- Confusing curved faces for flat faces—remember, cylinders and spheres have curved surfaces!
- Mixing up surface area and volume formulas—surface area (measured in sq units) and volume (in cubic units).
- Forgetting to include all faces when calculating total surface area (especially with complex solids).
Relation to Other Concepts
Three dimensional shapes are deeply related to two dimensional shapes (their faces are made up of 2D shapes). Mastering this topic also helps with understanding coordinates (coordinate geometry), visualising objects in space, and working with area and volume problems in later classes.
Classroom Tip
A simple way to remember the difference between 2D and 3D shapes: if you can pick it up and put something inside it (like a box or a cup), it’s three dimensional! Vedantu teachers recommend drawing and folding paper nets to physically see faces, edges, and vertices of each 3D solid.
We explored three dimensional shapes—from definition, types, properties, formulas, worked example, and practical tricks. Practise regularly and you’ll become confident at visualising and solving any problem on three dimensional shapes!
Cube and Cuboid Explained | Formulas and Volume Practice | Practice Worksheets
FAQs on Three Dimensional Shapes Explained: Types, Formulas & Uses
1. What are three-dimensional shapes?
Three-dimensional shapes, also known as 3D shapes or solid figures, are objects that have three dimensions: length, width, and height. Unlike two-dimensional shapes, they occupy space and have volume. Common examples include cubes, cuboids, spheres, cones, cylinders, and pyramids.
2. What are some examples of three-dimensional shapes in everyday life?
Many everyday objects are 3D shapes! Think of a cube (dice, Rubik's Cube), a cuboid (boxes, books), a sphere (balls, globes), a cylinder (cans, bottles), a cone (ice cream cones, party hats), and a pyramid (the Egyptian pyramids, some tents).
3. What is the difference between a cube and a cuboid?
Both cubes and cuboids are three-dimensional shapes with six faces. However, a cube has six identical square faces, while a cuboid has six rectangular faces (some may be squares). A cube is a special type of cuboid.
4. How do I calculate the volume of a cube?
The volume of a cube is calculated by cubing the length of one side (edge). The formula is: Volume = side × side × side = side³
5. What is the formula for the volume of a cylinder?
The volume of a cylinder is calculated using the formula: Volume = π × radius² × height, where π (pi) is approximately 3.14.
6. How do I find the surface area of a cuboid?
The surface area of a cuboid is found using the formula: Surface Area = 2(length × width + width × height + length × height)
7. What is the surface area of a sphere?
The surface area of a sphere is calculated using: Surface Area = 4 × π × radius²
8. What are the properties of a cone?
A cone has one circular base, one curved surface, and one vertex (point) at the top. Key properties include its radius (distance from the center of the base to any point on the circumference), height (perpendicular distance from the vertex to the base), and slant height (distance from the vertex to any point on the circumference of the base).
9. What is a net of a three-dimensional shape?
A net is a two-dimensional pattern that can be folded to form a three-dimensional shape. It shows all the faces of the 3D shape laid out flat.
10. What is the difference between 2D and 3D shapes?
2D shapes (two-dimensional) only have length and width (e.g., squares, circles). 3D shapes (three-dimensional) have length, width, and height, and occupy volume (e.g., cubes, spheres).
11. How many faces does a triangular prism have?
A triangular prism has five faces: two triangular bases and three rectangular lateral faces.
12. What are Platonic solids?
Platonic solids are convex regular polyhedra. This means they are three-dimensional shapes with all faces being congruent regular polygons, and the same number of faces meeting at each vertex. There are only five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
