Unit 5: Area and Volume (क्षेत्रफल र आयतन) Formulae
Introduction
Unit 5: Area and Volume (क्षेत्रफल र आयतन) covers the measurement of geometric figures. This chapter includes formulae for the area and perimeter of plane figures (Triangles, Quadrilaterals) and the volume and surface area of 3D solids (Prisms, Cylinders, Spheres, Cones, Pyramids).
Key Notations:
- $A$: Area (क्षेत्रफल)
- $P$: Perimeter (परिधि)
- $V$: Volume (आयतन)
- $S$: Curved Surface Area
- $l$: Slant Height (छड्के उचाइ)
- $h$: Vertical Height (ठाडो उचाइ)
- $r$: Radius (अर्धव्यास)
- $d$: Diameter (व्यास)
1. Plane Surfaces (समतल सतहहरू)
| 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) |
$$A = \frac{1}{2} p \times b$$ | $$P = p + b + h$$ |
|
Parallelogram (समानान्तर चतुर्भुज) |
$$A = b \times h$$ | $$P = 2(l + b)$$ |
|
Rhombus (समबाहु चतुर्भुज) |
$$A = \frac{1}{2} d_1 \times d_2$$ | $$P = 4l$$ |
|
Trapezium (समलम्ब चतुर्भुज) |
$$A = \frac{1}{2} h (a + b)$$ | $$P = Sum\ of\ all\ sides$$ |
|
Kite / Arrowhead (चङ्गा / एरोहेड) |
$$A = \frac{1}{2} d_1 \times d_2$$ | $$P = 2(a + b)$$ |
|
Quadrilateral (चतुर्भुज) |
$$A = \frac{1}{2} d(p_1 + p_2)$$ | $$P = Sum\ of\ all\ sides$$ |
2. Solids (ठोस वस्तुहरू)
A. Prism (प्रिज्मीय सतह)
| Lateral/Curved Surface Area (LSA) | $$LSA = Perimeter \times height = P \times h$$ |
| Total Surface Area (TSA) | $$TSA = LSA + 2 \times Base\ Area$$ |
| Volume (V) | $$V = Base\ Area \times height = A \times h$$ |
B. Cylinder (बेलना)
| Volume (V) | $$V = \pi r^2 h$$ |
| Curved Surface Area (CSA) | $$CSA = 2 \pi r h$$ |
| Total Surface Area (TSA) | $$TSA = 2 \pi r (r + h)$$ |
C. Sphere & Hemisphere (गोला र अर्धगोला)
| Solid | Volume (V) | Surface Area (A) |
|---|---|---|
| Sphere (गोला) |
$$V = \frac{4}{3} \pi r^3$$ | $$A = 4 \pi r^2$$ |
| Hemisphere (अर्धगोला) |
$$V = \frac{2}{3} \pi r^3$$ |
CSA: $$2 \pi r^2$$ TSA: $$3 \pi r^2$$ |
3. Cone and Pyramid (सोली र पिरामिड)
A. Cone (सोली)
[Image of cone geometry formulas]Slant height (छड्के उचाइ): $$l = \sqrt{h^2 + r^2}$$
Volume
$$V = \frac{1}{3} \pi r^2 h$$
Curved Surface Area
$$S = \pi r l$$
Total Surface Area
$$A = \pi r (r + l)$$
B. Pyramid (पिरामिड) – Square Base
Where $a$ = base side, $h$ = vertical height, $l$ = slant height.
Relation: $$l^2 = h^2 + (a/2)^2$$
| Volume (V) | $$V = \frac{1}{3} \times Base\ Area \times h = \frac{1}{3} a^2 h$$ |
| Lateral Surface Area (LSA) | $$LSA = \frac{1}{2} P \times l = 2al$$ |
| Total Surface Area (TSA) | $$TSA = a^2 + 2al$$ |
4. Combined Solids (संयुक्त ठोस वस्तुहरू)
| Combination | Volume (आयतन) | Curved Area (वक्र सतह) | Total Area (पूरा सतह) |
|---|---|---|---|
| Cylinder + Cone (Tent/ Pencil) |
$$V = \pi r^2 (h_1 + \frac{h_2}{3})$$ | $$S = \pi r (l + 2h_1)$$ | $$A = \pi r (l + r + 2h_1)$$ |
| Cylinder + Hemisphere (Test Tube/ Silo) |
$$V = \pi r^2 (h + \frac{2}{3}r)$$ | $$S = \pi r (2h + 2r)$$ | $$A = \pi r (2h + 3r)$$ |
| Cone + Hemisphere (Ice-cream/ Top) |
$$V = \frac{1}{3} \pi r^2 (h + 2r)$$ | $$S = \pi r (l + 2r)$$ | $$A = \pi r (l + 2r)$$ |
| Cylinder + 2 Hemispheres (Capsule) |
$$V = \pi r^2 (h + \frac{4}{3}r)$$ | $$S = 2 \pi r (h + 2r)$$ | $$A = 2 \pi r (h + 2r)$$ |
Full Chapter PDF Manual
For offline study, access the complete PDF manual below.
Disclaimer: This content is based on the Class 10 Compulsory Mathematics manual by Dr. Simkhada (Readmore Publishers). Please refer to the PDF for diagrammatic representations.