Definition Properties and Formulas of Three Dimensional Solids with Examples
The concept of Solids in Maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Solids in Maths?
Solids in Maths are three-dimensional (3D) shapes that have length, breadth, and height. Unlike 2D figures (which only have length and breadth), solids occupy space and have volume. Classic examples include the cube, cuboid, sphere, cylinder, cone, prism, and pyramid. You’ll find this concept applied in areas such as geometry solids, calculating volume and area, and real-life measurement problems. If you look around, you will see solids in everyday objects: a gift box (cuboid), a cricket ball (sphere), or a water tank (cylinder).
Why Study Solids in Maths?
Understanding solid shapes helps students visualize 3D objects, solve measurement questions in exams, and connect mathematical knowledge with real-world usage. This knowledge is essential for board exams, Olympiads, and practical life (like packing, filling, or building).
Types of Solids in Maths
- Cube – All sides and faces equal
- Cuboid – Like a box, faces are rectangles
- Sphere – Perfectly round, like a ball
- Cylinder – Two circular faces, one curved surface (like a can)
- Cone – Circular base, tapers to a point (like an ice-cream cone)
- Prism – Ends are parallel polygons, sides are rectangles
- Pyramid – Polygonal base, sides are triangles meeting at a point
Properties of Solid Shapes
| Solid Shape | Faces | Edges | Vertices | Example |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | Dice |
| Cuboid | 6 | 12 | 8 | Book, brick |
| Sphere | 1 curved | 0 | 0 | Football |
| Cylinder | 2 flat, 1 curved | 2 | 0 | Cola can |
| Cone | 1 flat, 1 curved | 1 | 1 | Ice-cream cone |
| Prism (Triangular) | 5 | 9 | 6 | Tent |
| Square Pyramid | 5 | 8 | 5 | Pyramid of Giza |
Key Formulas for Solids in Maths
Below are the main formulas for volume and surface area of some common solids:
| Solid | Volume (V) | Total Surface Area (TSA) |
|---|---|---|
| Cube (side = a) | \( a^3 \) | \( 6a^2 \) |
| Cuboid (l, b, h) | \( l \times b \times h \) | \( 2(lb + lh + bh) \) |
| Sphere (radius = r) | \( \frac{4}{3} \pi r^3 \) | \( 4\pi r^2 \) |
| Cylinder (r, h) | \( \pi r^2 h \) | \( 2\pi r (r + h) \) |
| Cone (r, h, l) | \( \frac{1}{3}\pi r^2 h \) | \( \pi r (l + r) \) |
Tip: Always keep a formula sheet handy, like the ones available on Vedantu’s Mensuration Formulas page!
Everyday Examples of Solids
| Real-Life Object | Shape in Maths |
|---|---|
| Ice cube, dice | Cube |
| Shoebox, brick | Cuboid |
| Football, orange | Sphere |
| Cola can, water tank | Cylinder |
| Ice-cream cone | Cone |
| Tent | Triangular prism |
Spotting these around you makes the concept of solids in maths easier and more memorable.
Step-by-Step Example: Volume of a Cuboid
Let’s solve:
Find the volume and surface area of a cuboid with length = 10 cm, breadth = 5 cm, height = 3 cm.
1. Write the formula for volume: \( V = l \times b \times h \ )2. Substitute the given values: \( V = 10 \times 5 \times 3 = 150 \text{ cm}^3 \)
3. Formula for total surface area: \( 2(lb + bh + lh) \)
4. Substitute the values: \( 2(10 \times 5 + 5 \times 3 + 10 \times 3) = 2(50 + 15 + 30) = 2 \times 95 = 190 \text{ cm}^2 \)
So, the cuboid has a volume of 150 cm³ and a surface area of 190 cm².
Speed Trick: Remembering Cube Numbers
To find the cube of small numbers fast (like 8³ or 11³):
1. 8³ = 512 (just 8 × 8 = 64, then 64 × 8 = 512)2. 11³ = (10 + 1)³ = 10³ + 3 × 10² × 1 + 3 × 10 × 1² + 1³ = 1000 + 300 + 30 + 1 = 1331
Practicing break-down steps saves time in the exam!
Try These Yourself
- What is the volume of a cube of side 6 cm?
- Write the number of faces, edges, and vertices in a cylinder.
- Find the curved surface area of a cylinder with r = 7 cm, h = 10 cm.
- Identify 3 solid shapes in your classroom and name their types in maths.
Frequent Errors and Misunderstandings
- Confusing between area (2D) and surface area (3D) formulas.
- Using wrong height/radius values in cylinder/cone problems.
- Forgetting that sphere has only a curved surface, no faces/edges.
Relation to Other Concepts
Solids in Maths are linked with 3D shapes and properties, mensuration, and nets of solids. Understanding solids makes chapters on area, perimeter, and real-life measurement easier.
Classroom Tip
A helpful way to remember properties: “Face-Edge-Vertex Table.” Recite faces–edges–vertices for cube, cuboid, and prism as 6-12-8. Vedantu teachers use 3D models and digital quizzes for easy recall!
Interlinks for Further Learning
- Surface Area and Volume of Cuboid – Master formula-based questions.
- Properties of 3D Shapes – Learn faces, edges, vertices in detail.
- Types of Polyhedra and Prisms – Advanced study for class 9/10.
- Nets of Solid Shapes – For better visualization and net-based questions.
- Solid Geometry Concepts – For deeper theory and practical maths.
We explored Solids in Maths—from definition, types, formula, real-life connections, pitfalls, and linked concepts. With continued practice and Vedantu’s expert sessions, you’ll become confident in all solid shape problems!
FAQs on Solids in Geometry and Three Dimensional Shapes
1. What are solids in geometry?
In geometry, solids are three-dimensional (3D) shapes that have length, width, and height. Unlike 2D figures, solids occupy space and have volume.
- They are also called three-dimensional shapes or 3D figures.
- Common examples include cube, cuboid, sphere, cylinder, cone, and pyramid.
- Solids have properties such as faces, edges, vertices, surface area, and volume.
2. What is the difference between 2D shapes and 3D solids?
The main difference is that 2D shapes have only length and width, while 3D solids have length, width, and height.
- 2D shapes (like squares and circles) have area but no volume.
- 3D solids (like cubes and spheres) have both surface area and volume.
- Solids occupy space, while 2D shapes lie flat on a plane.
3. What are faces, edges, and vertices in solids?
In a solid, faces are flat surfaces, edges are line segments where two faces meet, and vertices are corner points where edges meet.
- A cube has 6 faces, 12 edges, and 8 vertices.
- A cylinder has 2 circular faces and 1 curved surface but no vertices.
- A sphere has no edges or vertices.
4. What is the formula for the volume of a cube?
The volume of a cube is given by the formula V = a³, where a is the length of one side.
- If side length = 4 cm, then volume = 4³ = 64 cm³.
- All edges of a cube are equal.
- Volume is measured in cubic units like cm³ or m³.
5. What is the formula for the volume of a cuboid?
The volume of a cuboid is V = l × b × h, where l is length, b is breadth, and h is height.
- Example: If l = 5 cm, b = 3 cm, h = 2 cm, then volume = 5 × 3 × 2 = 30 cm³.
- A cuboid has 6 rectangular faces.
- This formula calculates the space occupied by the solid.
6. What is the volume formula for a cylinder?
The volume of a cylinder is V = πr²h, where r is the radius of the base and h is the height.
- Example: If r = 3 cm and h = 5 cm, then volume = π × 3² × 5 = 45π cm³.
- The base of a cylinder is circular.
- Use π ≈ 3.14 for numerical answers.
7. What is the volume formula for a sphere?
The volume of a sphere is given by V = (4/3)πr³, where r is the radius.
- Example: If r = 2 cm, volume = (4/3)π × 8 = (32/3)π cm³.
- A sphere has no edges or vertices.
- All points on the surface are equidistant from the center.
8. What is the difference between surface area and volume of solids?
The surface area of a solid is the total area of its outer surfaces, while volume is the space it occupies.
- Surface area is measured in square units (cm², m²).
- Volume is measured in cubic units (cm³, m³).
- Example: A cube with side 3 cm has surface area = 6 × 3² = 54 cm² and volume = 27 cm³.
9. What is Euler’s formula for polyhedra?
Euler’s formula for polyhedra is V − E + F = 2, where V = vertices, E = edges, and F = faces.
- For a cube: 8 − 12 + 6 = 2.
- This formula applies to convex polyhedra.
- It helps verify the correctness of face, edge, and vertex counts.
10. What are some real-life examples of solids?
Common real-life examples of solids include objects that have three dimensions and occupy space.
- A dice is a cube.
- A brick is a cuboid.
- A can is a cylinder.
- A ball is a sphere.
- An ice cream cone is a cone.
