Quick Revision Notes: Class 10 Mathematics

Chapter: Surface Areas and Volumes

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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.

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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)

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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

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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

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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

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5. Sphere (गोला)

• Radius = $r$
• Surface Area (CSA/TSA): 4πr^2
• Volume: (4/3)πr^3

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6. Hemisphere (अर्धगोला)

• Radius = $r$
• Curved Surface Area (CSA): 2πr^2
• Total Surface Area (TSA): 3πr^2
• Volume: (2/3)πr^3

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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)

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Important Concepts for MP Board Exam

1. Conversion of Solids (ठोसों का रूपांतरण):

   • When a solid of one shape is converted into another shape, the total volume remains constant.
   • $\text{Volume of Solid 1} = \text{Volume of Solid 2}$
   • $\text{Number of new objects} = \frac{\text{Volume of original solid}}{\text{Volume of one new object}}$

2. 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.

3. Value of Pi ($\pi$):

   • Use $\pi = \frac{22}{7}$ or $3.14$ as specified in the question. If not specified, use $\frac{22}{7}$.