Geometry is an essential branch of mathematics that deals with shapes, sizes, relative positions of figures, and properties of space. In this chapter, we explore various geometrical shapes and solid figures along with their characteristics.

Types of Geometrical Shapes

2D Shapes

Two-dimensional (2D) shapes have only length and breadth. They lie on a plane and are also called flat shapes. Common 2D shapes include:

• Circle: A round shape with no corners or edges. Every point on its boundary is equidistant from the center.
• Square: A shape with four equal sides and four right angles.
• Rectangle: A shape with opposite sides equal and four right angles.
• Triangle: A shape with three sides and three angles. Types of triangles include:
  • Equilateral Triangle: All sides and angles are equal.
  • Isosceles Triangle: Two sides and two angles are equal.
  • Scalene Triangle: All sides and angles are different.
• Polygon: A closed shape with three or more straight sides, such as pentagon, hexagon, etc.

3D Shapes

Three-dimensional (3D) shapes have length, breadth, and height. They occupy space and are also called solid shapes. Common 3D shapes include:

• Cube: A solid with six square faces of equal size.
• Cuboid: A solid with six rectangular faces.
• Sphere: A round solid shape where every point on the surface is equidistant from the center.
• Cylinder: A solid with two parallel circular bases connected by a curved surface.
• Cone: A solid with a circular base and a pointed top.
• Pyramid: A solid with a polygonal base and triangular faces meeting at a point.

Properties of 2D and 3D Shapes

2D Shapes

Shape	Sides	Angles	Special Property
Circle	0	0	All points equidistant from the center
Square	4	90°	All sides equal
Rectangle	4	90°	Opposite sides equal
Triangle	3	Sum of angles = 180°	Varies with type

3D Shapes

Shape	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Sphere	1	0	0
Cylinder	3	2	0
Cone	2	1	1
Pyramid	Varies	Varies	Varies

Geometric Formulas

2D Shapes

• Perimeter of a Square: 4 × side
• Perimeter of a Rectangle: 2 × (length + breadth)
• Area of a Square: side × side
• Area of a Rectangle: length × breadth
• Area of a Circle: π × radius²
• Circumference of a Circle: 2 × π × radius
• Area of a Triangle: 1/2 × base × height

3D Shapes

• Surface Area of a Cube: 6 × side²
• Surface Area of a Cuboid: 2 × (lb + bh + hl)
• Volume of a Cube: side³
• Volume of a Cuboid: length × breadth × height
• Volume of a Sphere: (4/3) × π × radius³
• Volume of a Cylinder: π × radius² × height
• Volume of a Cone: (1/3) × π × radius² × height

Applications of Geometry

Geometry has practical applications in various fields such as architecture, engineering, art, and everyday activities. Understanding shapes and solids helps in designing objects, constructing buildings, and solving real-life problems.

Fun Facts

• A sphere has the smallest surface area for a given volume compared to other shapes.
• The Great Pyramid of Giza is an example of a geometric pyramid in architecture.

Practice Problems

1. Find the perimeter and area of a rectangle with length 8 cm and breadth 5 cm.

2. Calculate the volume of a cube with side 4 cm.

3. A cone has a radius of 3 cm and a height of 5 cm. Find its volume.

4. Determine the surface area of a sphere with a radius of 7 cm.

Conclusion

Geometrical shapes and solids form the foundation of understanding space and structure. By learning their properties, formulas, and applications, students can enhance their problem-solving skills and logical thinking.