Unit 10: Area of Triangles and Quadrilaterals (त्रिभुज र चतुर्भुजको क्षेत्रफल) Formulae
Introduction
Unit 10: Area of Triangles and Quadrilaterals (त्रिभुज र चतुर्भुजको क्षेत्रफल) focuses on calculating the area and perimeter of various 2D geometric shapes. This unit is fundamental for understanding geometry in the SEE curriculum.
Key Notations:
- $A$: Area (क्षेत्रफल)
- $P$: Perimeter (परिधि)
- $b$: Base (आधार)
- $h$: Height (उचाइ)
- $s$: Semi-perimeter (अर्ध-परिधि)
- $d_1, d_2$: Diagonals (विकर्णहरू)
Important Properties (महत्वपूर्ण गुणहरू)
- Parallelograms on the same base and between the same parallel lines are equal in area.
एउटै आधार र उही समानान्तर रेखाहरू बिचमा रहेका समानान्तर चतुर्भुजहरूको क्षेत्रफल बराबर हुन्छ । - The area of a triangle is half the area of a parallelogram standing on the same base and between the same parallel lines.
एउटै आधार र उही समानान्तर रेखाहरू बिचमा रहेका त्रिभुजको क्षेत्रफल समानान्तर चतुर्भुजको क्षेत्रफलको आधा हुन्छ । - Triangles on the same base and between the same parallel lines are equal in area.
एउटै आधार र उही समानान्तर रेखाहरू बिचमा रहेका त्रिभुजहरूको क्षेत्रफल बराबर हुन्छ । - Triangles with equal area standing on the same base are between the same parallel lines.
एउटै आधारमा बनेका र क्षेत्रफल बराबर भएका त्रिभुजहरू उही समानान्तर रेखाहरूको बिचमा पर्दछन् । - The median of a triangle bisects it into two triangles of equal area.
त्रिभुजको मध्यिकाले त्रिभुजलाई बराबर क्षेत्रफल भएका दुई त्रिभुजमा विभाजन गर्दछ ।
1. Triangles (त्रिभुजहरू)
| Figure (चित्र) | Area (क्षेत्रफल) | Perimeter (परिधि) |
|---|---|---|
|
General Triangle (त्रिभुज) |
$$A = \frac{1}{2}b \times h$$
$$A = \sqrt{s(s-a)(s-b)(s-c)}$$ |
$$P = a + b + c$$ $$s = \frac{a+b+c}{2}$$ |
|
Equilateral Triangle (समबाहु त्रिभुज) |
$$A = \frac{\sqrt{3}}{4}a^2$$ | $$P = 3a$$ |
|
Isosceles Triangle (समद्विबाहु त्रिभुज) |
$$A = \frac{b}{4}\sqrt{4a^2-b^2}$$ | $$P = 2a + b$$ |
|
Right Angled Triangle (समकोणी त्रिभुज) |
$$A = \frac{1}{2} b \times p$$ | $$P = p + b + h$$ $$h = \sqrt{p^2 + b^2}$$ |
2. Quadrilaterals (चतुर्भुजहरू)
| Figure (चित्र) | Area (क्षेत्रफल) | Perimeter (परिधि) |
|---|---|---|
|
Parallelogram (समानान्तर चतुर्भुज) |
$$A = b \times h$$ | $$P = 2(l + b)$$ or $2(AB + AD)$ |
|
Rhombus (समबाहु चतुर्भुज) |
$$A = \frac{1}{2} d_1 \times d_2$$ | $$P = 4a$$ |
|
Quadrilateral (चतुर्भुज) |
$$A = \frac{1}{2} d(p_1 + p_2)$$ | $$P = AB+BC+CD+DA$$ |
|
Trapezium (समलम्ब चतुर्भुज) |
$$A = \frac{1}{2} h (p_1 + p_2)$$ (Note: $p_1, p_2$ are parallel sides) |
$$P = Sum\ of\ all\ sides$$ |
|
Arrowhead (एरोहेड) |
$$A = \frac{1}{2} d_1 \times d_2$$ | $$P = AB+BC+CD+AD$$ |
|
Kite (चङ्गा) |
$$A = \frac{1}{2} d_1 \times d_2$$ | $$P = AB+BC+CD+DA$$ |
Full Chapter PDF Manual
For offline study, access the complete PDF manual below.
Disclaimer: This content is extracted from the Class 10 Compulsory Mathematics manual by Dr. Simkhada (Readmore Publishers). Please refer to the PDF for diagrammatic representations.