CBSE Class 10 Maths Notes: Surface Areas and Volumes

✨ Introduction: Surface Areas and Volumes

Welcome to the fascinating world of Surface Areas and Volumes! This chapter explores how to calculate the space occupied by 3D objects (volumes) and the total area of their surfaces. We’ll be diving into the properties of common shapes and learning how to solve complex, real-world problems.

🧱 Fundamental Shapes & Formulae

Let’s explore the basic 3D shapes and their essential properties. Understanding these will be the foundation for tackling more complex problems.

1. Cube

A cube is a 3D shape with six identical square faces.

• **Surface Area (SA):** $6a^2$, where ‘a’ is the side length.
• **Volume (V):** $a^3$

2. Cuboid

A cuboid is a 3D shape with six rectangular faces. Think of a box!

• **Surface Area (SA):** $2(lb + bh + lh)$, where ‘l’ is length, ‘b’ is breadth, and ‘h’ is height.
• **Volume (V):** $lbh$

3. Sphere

A sphere is a perfectly round 3D object, like a ball.

• **Surface Area (SA):** $4\pi r^2$, where ‘r’ is the radius.
• **Volume (V):** $\frac{4}{3}\pi r^3$

4. Hemisphere

A hemisphere is half a sphere.

• **Curved Surface Area (CSA):** $2\pi r^2$
• **Total Surface Area (TSA):** $3\pi r^2$
• **Volume (V):** $\frac{2}{3}\pi r^3$

5. Right Circular Cylinder

A cylinder has two circular bases and a curved surface connecting them.

• **Curved Surface Area (CSA):** $2\pi rh$, where ‘r’ is the radius and ‘h’ is the height.
• **Total Surface Area (TSA):** $2\pi r(h + r)$
• **Volume (V):** $\pi r^2h$

6. Right Circular Cone

A cone has a circular base and a vertex.

• **Curved Surface Area (CSA):** $\pi rl$, where ‘r’ is the radius and ‘l’ is the slant height ($\sqrt{r^2 + h^2}$).
• **Total Surface Area (TSA):** $\pi r(l + r)$
• **Volume (V):** $\frac{1}{3}\pi r^2h$

🛠️ Combinations of Solids

This section focuses on problems involving two or more shapes combined. The key is to break down the problem and identify the individual shapes and their respective formulae.

Example: Imagine a solid toy shaped like a cone mounted on a hemisphere. The toy’s total surface area is the sum of the cone’s curved surface area and the hemisphere’s curved surface area. The total volume is the sum of the cone’s volume and the hemisphere’s volume.

• Strategy: Identify the individual shapes. Calculate the necessary dimensions (radius, height, slant height). Use the correct formulas for SA and Volume of each shape. Combine the results.

🧐 Visualisation and Problem Solving

This section helps you visualize problems and choose the right formulas.

• 1. Problem Type: A solid is melted and recast into another shape.
• Approach: The volume remains constant. Equate the volumes of the initial and final shapes.
• Example: A metallic sphere is melted and reshaped into a cylinder. The volume of the sphere equals the volume of the cylinder.
• 2. Problem Type: Hollow objects.
• Approach: Calculate the volume of the material by subtracting the inner volume from the outer volume.
• Example: A hollow cylinder has a certain thickness. To find the volume of the material, subtract the volume of the inner cylinder from the volume of the outer cylinder. For surface area, consider the outer surface, inner surface, and the areas of the top and bottom rings.
• 3. Problem Type: Painting or Covering Surfaces.
• Approach: The relevant calculation is the surface area. Determine the area to be painted/covered and use the appropriate formula.
• Example: Finding the cost of painting a cylindrical pillar – use the curved surface area.