Geometry Basics (Shapes, Perimeter & Area)

Last Updated: 06-Sep-2025

Geometry is the study of shapes, sizes, and space.

Basic 2D Shapes

• Triangle → 3 sides
• Square → 4 equal sides
• Rectangle → Opposite sides equal
• Circle → Round, no sides

👉 Real Life:

• Triangle → Traffic sign
• Rectangle → Book, phone screen
• Circle → Coin, wheel

Perimeter (Boundary Length)

Perimeter = distance around a shape.

• Square → 4 x side
• Rectangle → 2 x (length + width)
• Triangle → sum of 3 sides
• Circle → $2 \pi r$ (r = radius)

👉 Example:

Square side = 5 cm → Perimeter = 4×5 = 20 cm.

Area (Space inside)

Area = region covered inside shape.

• Square → $s^2$
• Rectangle → $l\times w$
• Triangle → $\frac{1}{2} \times b \times h$
• Circle → $\pi r^2$

👉 Example:

Circle radius = 7 cm → Area $\approx 3.14 \times 7^2 = 153.86\text{ cm}^2$.

Angles

• Right angle = $90^\circ$
• Acute angle $< 90^\circ$
• Obtuse angle $> 90^\circ$ but $< 180^\circ$
• Straight angle = $180^\circ$

👉 Real Life:

• $90^\circ$ → Corner of a paper.
• $180^\circ$ → Flat line.

3D Shapes (Solid Geometry)

• Cube → Box with all equal sides
• Cuboid → Box (like brick)
• Sphere → Ball
• Cylinder → Pipe
• Cone → Ice-cream cone

👉 Extra formulas:

• Volume of cube = $s^3$
• Volume of cuboid = $l \times w \times h$
• Volume of sphere = $\frac{4}{3}\pi r^3$

Pythagoras Theorem

In a right-angled triangle,

$(\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Height})^2
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\text{Hypotenuse} = \sqrt{(\text{Base})^2 + (\text{Height})^2}$

👉 Example:

Base = 3, Height = 4 → Hypotenuse ?

$Hypotenuse = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$

Polygons

A polygon is a closed shape made of straight lines.

• Triangle → 3 sides
• Quadrilateral → 4 sides
• Pentagon → 5 sides
• Hexagon → 6 sides

👉 Formula:

$\text{Sum of interior angles} = (n - 2) \times 180^\circ$

• Triangle → $(3 - 2) \times 180^\circ = 180^\circ$
• Pentagon → $(5 - 2) \times 180^\circ = 540^\circ$

Circles

Parts of a circle:

• Radius ($r$): center to boundary
• Diameter ($d$): $2 \times$ radius
• Circumference ($C$): distance around circle $= 2\pi r$
• Chord: line joining two points on circle
• Arc: part of boundary
• Sector: slice of a circle (like pizza piece)

👉 Example:

Circle radius = 7 → Diameter = 14, Circumference $\approx 44$.

Symmetry

• A shape has line of symmetry if it can be folded into two equal halves.
• Example:
  • Square → 4 symmetry lines
  • Rectangle → 2 symmetry lines
  • Circle → infinite symmetry lines

Coordinate Geometry Basics

We use an x-axis (horizontal) and Y-axis (vertical) to locate points.

A point is written as (x,y).

Example:

Point (2,3) → move 2 right on x, 3 up on y.

👉 Distance between Points $(x_1,y_1)$ and $(x_2,y_2)$:

$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

👉 Example:

Distance between (0,0) and (3,4):

$\sqrt{3^2+4^2}=5$

(again Pythagoras!)

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