Three Dimensional Shapes - Properties Of 3D Shapes With Curves

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 × 3 = 12 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