What Are the Types and Properties of Solids in Maths?
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 Mathematics: Definition, Types, Properties & Formulas
1. What are solids in Maths?
In Maths, solids are three-dimensional (3D) shapes that have length, breadth, and height. Unlike two-dimensional shapes, solids occupy space. Common examples include cubes, cuboids, spheres, and cylinders. Understanding solids involves learning about their properties, such as faces, edges, and vertices, and calculating their volume and surface area.
2. What are the different types of solids in geometry?
Geometry includes various types of solids. Key examples are:
• Polyhedra (solids with flat faces): cubes, cuboids, prisms, pyramids
• Non-polyhedra (solids with curved surfaces): spheres, cylinders, cones. Each solid type has unique properties and formulas for calculating volume and surface area.
3. How do you find the volume and surface area of solids?
Calculating the volume and surface area of solids depends on their shape. Each type has a specific formula. For example:
• Cube: Volume = a³; Surface Area = 6a²
• Cuboid: Volume = l × b × h; Surface Area = 2(lb + bh + lh)
• Sphere: Volume = (4/3)πr³; Surface Area = 4πr²
Where 'a' is the side length, 'l', 'b', and 'h' are the length, breadth, and height of a cuboid, and 'r' is the radius of a sphere.
4. What are faces, edges, and vertices in solids?
Faces are the flat surfaces of a solid. Edges are the line segments where two faces meet. Vertices are the points where edges intersect (corners). For example, a cube has 6 faces, 12 edges, and 8 vertices.
5. What is Euler's formula for solids?
Euler's formula is a relationship between the number of faces (F), vertices (V), and edges (E) of a polyhedron: F + V - E = 2. This formula helps verify the structure of polyhedral solids.
6. How are solids related to real-world objects?
Many everyday objects are based on solid shapes. For example, a box is a cuboid, a ball is a sphere, and a can is a cylinder. Understanding solids helps us analyze and measure the volume and space occupied by these objects.
7. What are some common mistakes students make when working with solids?
Common errors include: using the wrong formula for a given solid, confusing surface area with volume, and incorrectly identifying the faces, edges, and vertices of complex solids. Careful attention to the shape's properties and the correct formula is crucial.
8. How can I improve my understanding of solids?
To master solids, practice solving various problems involving volume and surface area calculations. Use visual aids like 3D models and diagrams to improve spatial reasoning. Make sure to understand the different types of solids and their properties.
9. What are nets of solids?
A net is a two-dimensional pattern that can be folded to form a three-dimensional solid. Understanding nets helps visualize the relationship between a solid's surface and its unfolded form, aiding in surface area calculations.
10. What are the units used to measure the volume of solids?
Volume is measured in cubic units, such as cubic centimeters (cm³), cubic meters (m³), or cubic millimeters (mm³). The choice of unit depends on the size of the solid being measured.
11. What is the difference between a prism and a pyramid?
A prism has two identical parallel bases and rectangular lateral faces. A pyramid has one base and triangular lateral faces meeting at a single vertex (apex).
12. How do I remember the formulas for different solids?
Create flashcards or a chart summarizing the formulas for each solid type. Practice using the formulas in various problems. Relate the formulas to the properties of each solid to aid memorization. Consider using mnemonic devices to help remember complex formulas.
