Quick Revision Notes: Perimeter and Area

Class: 7 Mathematics (MP Board)
Chapter: Perimeter and Area

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1. Introduction

• Perimeter (परिमाप): The distance around a closed figure. It is the total length of the boundary. Measured in units like cm, m, km.
• Area (क्षेत्रफल): The measure of the region enclosed by a closed figure. Measured in square units like cm², m², km².

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2. Units of Conversion (महत्वपूर्ण इकाइयाँ)

• $1 \text{ metre (m)} = 100 \text{ centimetres (cm)}$
• $1 \text{ square metre (m}^2\text{)} = 100 \text{ cm} \times 100 \text{ cm} = 10,000 \text{ cm}^2$
• $1 \text{ hectare} = 10,000 \text{ m}^2$

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3. Formulas for Squares and Rectangles (वर्ग और आयत)

Rectangle (आयत)

• Let length $= l$ and breadth $= b$
• Perimeter ($P$) $= 2 \times (l + b)$
• Area ($A$) $= l \times b$
• Length ($l$) $=\frac{\text{Area}}{b}$ or $\frac{P}{2} - b$
• Breadth ($b$) $=\frac{\text{Area}}{l}$ or $\frac{P}{2} - l$
• Diagonal of a Rectangle $= \sqrt{l^2 + b^2}$

Square (वर्ग)

• Let the side of the square $= s$
• Perimeter ($P$) $= 4 \times s$
• Area ($A$) $= s \times s = s^2$
• Side ($s$) $=\frac{\text{Perimeter}}{4}$ or $\sqrt{\text{Area}}$

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4. Parallelogram and Triangle (समांतर चतुर्भुज और त्रिभुज)

Parallelogram (समांतर चतुर्भुज)

• Area $= \text{Base } (b) \times \text{Height } (h)$
• Note: Any side of a parallelogram can be chosen as a base. The perpendicular dropped on that side from the opposite vertex is known as the height (altitude).

Triangle (त्रिभुज)

• A triangle is half of a parallelogram.
• Area $=\frac{1}{2} \times \text{Base } (b) \times \text{Height } (h)$
• Note: All congruent triangles equal in area, but triangles with equal areas need not be congruent.

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5. Circles (वृत्त)

• Radius ($r$): Distance from the center to any point on the boundary.
• Diameter ($d$): A line segment passing through the center and joining two points on the circle ($d = 2r$).
• Pi ($\pi$): A constant approximately equal to $\frac{22}{7}$ or $3.14$.
• Circumference of a Circle ($C$) $= 2 \pi r$ (or $\pi d$)
• Area of a Circle ($A$) $= \pi r^2$

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6. Summary of Key Formulas at a Glance

Shape (आकृति)	Perimeter / Circumference (परिमाप / परिधि)	Area (क्षेत्रफल)
Rectangle	$2(l + b)$	$l \times b$
Square	$4s$	$s^2$
Parallelogram	$2(a + b)$ (where a, b are adjacent sides)	$b \times h$
Triangle	Sum of all three sides ($a + b + c$)	$\frac{1}{2} \times b \times h$
Circle	$2\pi r$	$\pi r^2$

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7. Important Tips for Problem Solving

1. Always check that all dimensions (length, breadth, radius, etc.) are in the same unit before calculating perimeter or area.
2. Remember to write the correct units in your final answer (e.g., cm or m for perimeter; $\text{cm}^2$ or $\text{m}^2$ for area).
3. To find the area of a pathway inside or outside a rectangle/square, calculate the area of the outer figure and subtract the area of the inner figure.