📝 Chapter Notes & Revision
Surface Areas and Volumes
📐 Formula & Cheat Sheet (English)
Quick Revision Notes: Class 10 Mathematics
Chapter: Surface Areas and Volumes
Introduction
In this chapter, we deal with the calculations of surface areas and volumes of combined solid figures, conversion of solids from one shape to another, and the frustum of a cone.
1. Cuboid (घनाभ)
A cuboid has 6 rectangular faces, 12 edges, and 8 vertices.
- Length = $l$, Breadth = $b$, Height = $h$
- Total Surface Area (TSA):
2(lb + bh + hl) - Lateral / Curved Surface Area (LSA/CSA):
2(l + b)h - Volume:
l × b × h - Length of Diagonal:
√(l^2 + b^2 + h^2)
2. Cube (घन)
A cube is a special cuboid where all dimensions are equal.
- Side = $a$
- Total Surface Area (TSA):
6a^2 - Lateral Surface Area (LSA):
4a^2 - Volume:
a^3 - Length of Diagonal:
a√3
3. Right Circular Cylinder (लंबवृतीय बेलन)
- Radius = $r$, Height = $h$
- Curved Surface Area (CSA):
2πrh - Total Surface Area (TSA):
2πr(r + h) - Volume:
πr^2h
4. Right Circular Cone (लंबवृतीय शंकु)
- Radius = $r$, Height = $h$, Slant Height = $l$
- Slant Height ($l$):
√(r^2 + h^2) - Curved Surface Area (CSA):
πrl - Total Surface Area (TSA):
πr(l + r) - Volume:
(1/3)πr^2h
5. Sphere (गोला)
- Radius = $r$
- Surface Area (CSA/TSA):
4πr^2 - Volume:
(4/3)πr^3
6. Hemisphere (अर्धगोला)
- Radius = $r$
- Curved Surface Area (CSA):
2πr^2 - Total Surface Area (TSA):
3πr^2 - Volume:
(2/3)πr^3
7. Frustum of a Cone (शंकु का छिन्नक)
When a cone is cut by a plane parallel to its base, the lower part is called the frustum.
- Radii of ends = $r_1$ and $r_2$ (where $r_1 > r_2$), Height = $h$, Slant Height = $l$
- Slant Height ($l$):
√(h^2 + (r_1 - r_2)^2) - Curved Surface Area (CSA):
π(r_1 + r_2)l - Total Surface Area (TSA):
π(r_1 + r_2)l + πr_1^2 + πr_2^2 - Volume:
(1/3)πh(r_1^2 + r_2^2 + r_1r_2)
Important Concepts for MP Board Exam
-
Conversion of Solids (ठोसों का रूपांतरण):
- When a solid of one shape is converted into another shape, the total volume remains constant.
- $\text{Volume of Solid 1} = \text{Volume of Solid 2}$
- $\text{Number of new objects} = \frac{\text{Volume of original solid}}{\text{Volume of one new object}}$
-
Combination of Solids (ठोसों का संयोजन):
- To find the TSA of a combined solid (e.g., a tent made of a cylinder and a cone, or a toy made of a hemisphere and a cone), add the individual visible curved surface areas.
- Note: $\text{TSA} \neq \text{Sum of individual TSAs}$ because internal intersecting areas are not exposed.
-
Value of Pi ($\pi$):
- Use $\pi = \frac{22}{7}$ or $3.14$ as specified in the question. If not specified, use $\frac{22}{7}$.