Solids

Introduction

In mathematics, solid shapes are three-dimensional figures that have length, width, and height. Unlike flat figures in 2D geometry, solid shapes have volume and occupy space. They are all around us: boxes, balls, cones, and cylinders are everyday examples. Understanding them and their properties helps us classify and work with them in both maths and real life.

 

Table of Contents

• What Are Solid Shapes?
• Properties of Solid Shapes
• Types of Solid Shapes
• Solved Examples
• Conclusion
• FAQs on Solid Shapes

 

What Are Solid Shapes?

Solid shapes are shapes that have three dimensions - length, width, and height. They are also called 3D shapes because they are not flat like 2D shapes (such as squares and circles). Solid shapes take up space and can be touched and held.

 

Properties of Solid Shapes

• Three Dimensions

• Solid shapes possess length, breadth (width), and height.

• This renders them 3D objects that can be quantified in cubic units.

• Volume

• Volume is the amount of space a solid takes up.

• Measured in cubic units (cm³, m³, etc.).

• Surface Area

• The overall area covering the solid, including all faces and curved surfaces.

• Measured in square units (cm², m², etc.).

• Faces

• Flat or curved surfaces of a solid.

• Example: A cube has 6 square faces; a sphere has 1 curved face.

• Edges

• Line segments where two faces meet.

• Example: A cuboid contains 12 edges.

• Vertices

• Points of intersection of edges.

• Example: A cube contains 8 vertices.

These characteristics of solid shapes are utilized to recognize and distinguish among various kinds of solids shapes.

 

Types of Solid Shapes

Some of the types of solids shapes are:

Cube

A cube is a 3D solid figure in which all the faces are congruent squares and all the edges have the same length. It is a regular polyhedron since all the faces, edges, and angles are equal.

Properties

• Faces - A cube consists of 6 faces, and every face is a square.

• Edges - A cube consists of 12 edges with equal length.

• Vertices - A cube has 8 vertices (corners) where three edges intersect.

• Angles - All angles between edges are 90°.

• Symmetry - A cube has 9 planes of symmetry and rotational symmetry.
  

Formulas for a Cube

• Lateral Surface Area (LSA) = 4 × a²

• Total Surface Area (TSA) = 6 × a²

• Volume (V) = a³

• Where:
  a = length of an edge of the cube

 

Cuboid

A cuboid is a three-dimensional (3D) solid figure with six rectangular faces. Opposite faces of a cuboid are equal and all its angles are right angles (90°). It is also known as a rectangular prism since its faces are rectangles.

Properties

• Faces - A cuboid has 6 faces, each a rectangle.

• Edges - A cuboid has 12 edges. Opposite edges are equal in length.

• Vertices - A cuboid has 8 vertices (corners).

• Angles - All angles are 90°.

• Shape of Faces - All faces are rectangles.

• Symmetry - Has plane and rotational symmetry along certain axes.

Formulas for a Cuboid

• Lateral Surface Area (LSA) = 2 × h × (l + b)

• Total Surface Area (TSA) = 2 × (l × b + b × h + h × l)

• Volume (V) = l × b × h

Where:

• l = length
• b = breadth (width)
• h = height

 

Cylinder

A cylinder is a three-dimensional solid with two parallel congruent circular bases joined by a curved surface. Sides are not plane but curved, and it differs from cubes or cuboids.

It is a solid figure in geometry.

Properties of a Cylinder

• Faces: 3 (2 circular bases + 1 curved surface)

• Edges: 2 (circumferences of the bases)

• Vertices: 0 (no sharp corners)

• Shape of bases: Circular

• Symmetry: Infinitely many lines of symmetry along the central axis.

Formulas for Cylinder

• Curved Surface Area (CSA) = 2 × π × r × h

• Total Surface Area (TSA) = 2 × π × r × h + 2 × π × r²

• Volume (V) = π × r² × h

Where:

• r = radius of the base
• h = height of the cylinder

 

Cone

A cone is a three-dimensional (3D) solid shape with one circular flat face (known as the base) and one curved surface that tapers smoothly to a single point known as the apex or vertex.

Properties of a Cone

• It has 1 circular base.

• It has 1 curved surface.

• It has 1 apex (vertex).

• It has 1 boundary (circular edge).

• It is not a polyhedron as it possesses a curved surface.

• Volume and surface area are radius and height dependent.

Formula of a Cone

• Curved Surface Area (CSA) = π × r × l

• Total Surface Area (TSA) = π × r × (r + l)

• Volume = (1/3) × π × r² × h

Where:

r = radius of the base

h = height of the cylinder

 

Sphere

A sphere is a completely round three-dimensional solid shape, the same as a ball. All points on the surface of a sphere are equidistant from its center. This is known as the radius of the sphere. A sphere definition in geometry is that it is a solid with no edges, no vertices, and no flat faces.

Properties

• Faces: 0 flat faces (1 continuous curved surface)

• Edges: 0

• Vertices: 0

• Each diameter is of the same length 2r

• Any cross-section passing through the center is a great circle (a circle with radius r).

• Surface is smooth, no edges, no corners.

Formulas for Sphere

• Curved Surface Area (CSA) / Total Surface Area (TSA) = 4 × π × r²

• Volume (V) = (4/3) × π × r³

Where:

r = radius of the sphere

 

Solved Examples

Example 1

Q: A cube has edge a = 4 cm. Find LSA, TSA and Volume.

Formulas: LSA = 4a², TSA = 6a², Volume = a³

Solution:

LSA = 4 × 4² = 4 × 16 = 64 cm²

TSA = 6 × 4² = 6 × 16 = 96 cm²

Volume = 4³ = 64 cm³

Answer: LSA = 64 cm², TSA = 96 cm², Volume = 64 cm³

 

Example 2

Q: A cylinder has radius r = 2 cm and height h = 7 cm. Find CSA, TSA and Volume. (Use π)

Formulas: CSA = 2πrh, TSA = 2πr(h + r), Volume = πr²h

Solution:

CSA = 2 × π × 2 × 7 = 28π cm²

TSA = 2 × π × 2 × (7 + 2) = 4π × 9 = 36π cm²

Volume = π × 2² × 7 = π × 4 × 7 = 28π cm³

Answer: CSA = 28π cm², TSA = 36π cm², Volume = 28π cm³

 

Example 3

Q: A cone with radius r = 3 cm and height h = 4 cm. Determine slant height l, CSA, TSA and Volume.

Formulas: l = √(r² + h²), CSA = πrl, TSA = πr(l + r), Volume = (1/3)πr²h

Solution:

l = √(3² + 4²) = √(9 + 16) = √25 = 5 cm

CSA = π × 3 × 5 = 15π cm²

TSA = π × 3 × (5 + 3) = 3π × 8 = 24π cm²

Volume = (1/3)π × 3² × 4 = (1/3)π × 9 × 4 = 12π cm³

Answer: l = 5 cm, CSA = 15π cm², TSA = 24π cm², Volume = 12π cm³

 

Example 4

Q: A cuboid has l = 6 cm, b = 4 cm, h = 3 cm. Find the space diagonal d.

Formula: d = √(l² + b² + h²)

Solution:

d = √(6² + 4² + 3²) = √(36 + 16 + 9) = √61 ≈ 7.81 cm

Answer: d = √61 cm (≈ 7.81 cm)

 

Example 5

Q: A closed cylindrical tank with radius r = 1.5 m and height h = 4 m. Find the area to paint (TSA).

Formula: TSA = 2πr(h + r)

Solution:

TSA = 2π × 1.5 × (4 + 1.5) = 3π × 5.5 = 16.5π m² ≈ 51.84 m²

Answer: Area to paint = 16.5π m² (≈ 51.84 m²)

 

Conclusion

Solids are a key part of geometry, combining length, width, and height to form 3D objects. By understanding the solid definition, properties of solid shapes, types of solid shapes, and solid shapes examples, we can connect theory to practical applications. Mastery of solid formulas and problem-solving enhances both mathematical knowledge and real-life problem-handling skills.

 

Frequently Asked Questions On Solid Shapes

1. What are solid shapes?

Answer: Solid shapes are three-dimensional (3D) figures that have length, width (breadth), and height. They occupy space, have volume, and can have flat or curved surfaces. Examples include cubes, cuboids, cones, cylinders, and spheres.

2. What are the 6 examples of solid shapes?

Answer: Six common examples of solid shapes are:

1. Cube

2. Cuboid

3. Cylinder

4. Cone

5. Sphere

6. Pyramid
   
   

3. Is a cylinder a solid shape?

Answer: Yes, a cylinder is a solid shape because it has three dimensions - length, width (or diameter), and height. That means it takes up space and is not flat like a 2D shape.

 

4. What are faces, edges, and vertices of a solid?

Answer:

• Faces: Flat or curved surfaces of a solid. Example: A cube has 6 faces.

• Edges: Straight lines where two faces meet. Example: A cuboid has 12 edges.

• Vertices: Points where edges meet. Example: A cube has 8 vertices.
  

 

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