Classwise Concept with Examples
6th	7th	8th	9th	10th	11th	12th

Class 9th Chapters
1. Number Systems	2. Polynomials	3. Coordinate Geometry
4. Linear Equations In Two Variables	5. Introduction To Euclid’s Geometry	6. Lines And Angles
7. Triangles	8. Quadrilaterals	9. Areas Of Parallelograms And Triangles
10. Circles	11. Constructions	12. Heron’s Formula
13. Surface Areas And Volumes	14. Statistics	15. Probability

Content On This Page
Basis Terms Realted to Solid Figures	Surface Area of Solid Figures	Volume of Solid Figures

Chapter 13 Surface Areas And Volumes (Concepts)

Welcome to this significantly expanded chapter on Mensuration, where we take a major leap forward from the concepts explored in Class 8. While previously we focused on the perimeter and area of 2D shapes and the basics of surface area and volume for cuboids, cubes, and cylinders, this chapter delves deeper into the three-dimensional world. We will introduce methods for calculating the Surface Areas and Volumes of more complex and commonly encountered solid shapes: the cone and the sphere, alongside a thorough review and application of the formulas for the solids already studied. Our goal is to develop a comprehensive understanding of how to quantify both the exterior surface and the space occupied by these fundamental 3D forms.

Before venturing into new territory, let's consolidate our understanding of the solids introduced earlier. Recall the key measurements: Lateral Surface Area (LSA) or Curved Surface Area (CSA), which measures the area of the side surfaces excluding bases; Total Surface Area (TSA), encompassing the area of all surfaces including bases; and Volume (V), representing the total space enclosed by the solid. We revisit the formulas for:

Cuboid (length $l$, breadth $b$, height $h$):
- $ LSA = 2h(l+b) $
- $ TSA = 2(lb + bh + hl) $
- $ V = l \times b \times h $
Cube (side $a$):
- $ LSA = 4a^2 $
- $ TSA = 6a^2 $
- $ V = a^3 $
Right Circular Cylinder (radius $r$, height $h$):
- $ CSA = 2\pi rh $
- $ TSA = 2\pi r(r + h) $
- $ V = \pi r^2 h $

Now, we introduce new shapes and their corresponding formulas. For the Right Circular Cone, we introduce the concept of slant height ($l$), which is the distance from the apex to any point on the circumference of the base. It relates to the radius ($r$) and perpendicular height ($h$) via the Pythagorean theorem: $l^2 = r^2 + h^2$. The formulas are:

$ CSA = \pi r l $
$ TSA = \pi r (l + r) $ (CSA plus area of the circular base, $\pi r^2$)
$ V = \frac{1}{3}\pi r^2 h $ (Crucially, this volume is exactly one-third that of a cylinder with the same base radius and height).

Next, we explore the perfectly symmetrical Sphere:

Surface Area = $ 4\pi r^2 $ (Note: A sphere doesn't have separate LSA/TSA; it has only one continuous surface).
Volume = $ \frac{4}{3}\pi r^3 $

We also consider the Hemisphere (exactly half a sphere):

$ CSA = 2\pi r^2 $ (Half the surface area of the sphere)
$ TSA = 3\pi r^2 $ (The CSA plus the area of the flat circular base, $\pi r^2$)
Volume = $ \frac{2}{3}\pi r^3 $ (Half the volume of the sphere)

Beyond calculating these quantities for individual, isolated shapes using their dimensions ($r, h, l, a, \dots$), a significant focus of this chapter involves tackling problems concerning combinations of solids. This requires strong spatial visualization skills. Examples include finding the surface area of a toy shaped like a cone mounted on a hemisphere, or the volume of a capsule formed by a cylinder with hemispherical ends. Solving these problems involves carefully identifying which surfaces are exposed (for TSA) or which volumes contribute to the total, often requiring adding or subtracting the respective calculated parts. Furthermore, numerous practical application problems are explored, such as determining the amount of canvas needed for a tent (surface area), finding the capacity of tanks or vessels (volume), calculating the costs (perhaps in $\textsf{₹}$) associated with painting surfaces or filling containers, and analyzing scenarios involving the conversion of a solid from one shape into another (by equating their volumes). Throughout all calculations, maintaining accuracy with units (e.g., $cm^2, m^2$ for area; $cm^3, m^3$, litres for volume) and performing necessary conversions remains critically important.

Basic Terms Related to Solid Figures

In earlier classes, our focus was primarily on two-dimensional (2D) geometric figures, such as squares, rectangles, triangles, and circles. These figures lie flat on a plane and are described by two dimensions, typically length and breadth. We learned how to calculate their perimeters (the length of their boundary) and areas (the measure of the surface enclosed within their boundary).

In this chapter, we move into the realm of three-dimensional (3D) geometry. We will study shapes that occupy space, known as solid figures or simply solids. These figures have three dimensions: length, breadth (often called width), and height (or depth). Common examples of solid figures encountered in everyday life include cuboids (like bricks or boxes), cubes (like dice), cylinders (like pipes or soda cans), cones (like ice cream cones), and spheres (like balls).

For solid figures, the concepts of perimeter and area of 2D shapes are extended and replaced by notions of surface area (the total area of all the exterior surfaces of the solid) and volume (the amount of space enclosed or occupied by the solid).

Key Concepts and Terminology

Solid Figure (or Solid):

A solid figure, or solid, is a geometric figure that exists in three dimensions and occupies a definite amount of space. It has length, breadth, and height, and it possesses volume and surface area.

Dimensions:

Three-dimensional figures are characterized by three fundamental measurements that define their size and extent in space: length, breadth (or width), and height (or depth). For example, a cuboid is defined by its length, breadth, and height.

Surface:

The surface of a solid is the exterior boundary that separates the solid from the surrounding space. A solid's surface can consist of flat parts (called plane faces) or curved parts (called curved surfaces).

Edge:

An edge is the line segment where two faces of a solid meet. Edges are typically straight in solids with flat faces.

Vertex:

A vertex (plural: vertices) is a point where three or more edges of a solid meet. Vertices are the "corners" of a solid.

Common Solid Figures and Their Components

Let's look at some common solid figures and identify their components:

Cuboid:

A cuboid is a solid figure bounded by six rectangular faces. It is a type of rectangular prism. In a cuboid, opposite faces are congruent (identical in shape and size). A cuboid has 12 edges and 8 vertices.

Example: A matchbox, a brick, a rectangular room.

Components: 6 rectangular faces, 12 edges, 8 vertices.

Cube:

A cube is a special type of cuboid in which all edges are of equal length. Consequently, all six faces of a cube are congruent squares. A cube also has 12 edges and 8 vertices.

Example: A dice, a sugar cube, a Rubik's cube.

Components: 6 square faces, 12 edges, 8 vertices.

Cylinder:

A cylinder is a solid figure with two congruent circular bases that are parallel to each other, connected by a curved lateral surface. The axis of the cylinder is the line segment joining the centres of the two bases.

Example: A can of soft drink, a circular pillar, a section of pipe.

Components: Two circular plane bases, one curved lateral surface.

Cone:

A cone is a solid figure with a single circular base and a curved lateral surface that tapers smoothly from the base to a point called the apex or vertex. The height of the cone is the perpendicular distance from the apex to the centre of the base. The slant height is the distance from the apex to any point on the circumference of the base.

Example: An ice-cream cone, a birthday party hat, a traffic cone.

Components: One circular plane base, one curved lateral surface, an apex (vertex).

Sphere:

A sphere is a perfectly round solid figure in three-dimensional space. It is defined as the set of all points in space that are at a fixed distance from a fixed central point. It has no edges or vertices.

Example: A ball, a globe, a marble.

Components: One continuous curved surface.

In the following sections, we will learn how to calculate the surface areas and volumes of these common solid figures, along with applying these calculations to solve real-world problems.

Surface Area of Solid Figures

The surface area of a three-dimensional solid figure is the total area of all the surfaces (or faces) that enclose the solid. Imagine unfolding the solid and measuring the area of each piece; the sum of these areas is the surface area. Surface area is a measure of the extent of the boundary of a solid and is always expressed in square units (e.g., $\text{cm}^2$, $\text{m}^2$).

Types of Surface Area

For many solid figures, particularly those with distinct bases, we distinguish between two types of surface area:

Total Surface Area (TSA): This is the sum of the areas of all the faces (including the top and bottom bases, if any) that form the surface of the solid.
Lateral Surface Area (LSA) or Curved Surface Area (CSA): This is the sum of the areas of the lateral faces (the sides) of the solid, excluding the area of the top and bottom bases. The term 'Curved Surface Area' is used specifically for solids that have a curved side surface, such as cylinders and cones, while 'Lateral Surface Area' is often used for polyhedrons like cuboids and cubes.

Surface Area Formulas for Common Solids

The surface area formulas depend on the shape of the solid and its dimensions. These formulas are derived by calculating the area of each individual face or curved surface and summing them up.