Surface Areas and Volumes Class 9 ||Maths|| Chapter 11 NCERT Notes

Surface Areas and Volumes Class 9 ||Maths|| Chapter 11 NCERT Notes

Surface Areas and Volumes ||Maths|| Chapter 11 NCERT Notes

1. Cuboid

A cuboid is a 3D object with six rectangular faces, where opposite faces are equal.
Surface Area:
Total Surface Area (TSA) of a cuboid = Sum of the areas of all six rectangular faces.
$TSA of cuboid = 2 (l b + b h + h l)$
Where $l$ is the length, $b$ is the breadth, and $h$ is the height of the cuboid.
Lateral Surface Area (LSA): The area of all the faces excluding the top and bottom.
$LSA of cuboid = 2 h (l + b)$
Volume:
Volume of a cuboid is the amount of space enclosed by the cuboid.
$Volume of cuboid = l \times b \times h$

2. Cube

A cube is a special case of a cuboid where all the sides are equal, i.e., $l = b = h = a$ .
Surface Area:
Total Surface Area (TSA) of a cube = Sum of the areas of all six square faces.
$TSA of cube = 6 a^{2}$
Lateral Surface Area (LSA): The area of all the faces excluding the top and bottom.
$LSA of cube = 4 a^{2}$
Volume:
Volume of a cube is the amount of space enclosed by the cube.
$Volume of cube = a^{3}$

3. Right Circular Cylinder

A cylinder is a solid object with two parallel circular bases connected by a curved surface. The line joining the centers of the bases is called the height $h$ .
Surface Area:
Curved Surface Area (CSA): The area of the curved surface connecting the two circular bases.
$CSA of cylinder = 2 π r h$
Where $r$ is the radius of the base and $h$ is the height of the cylinder.
Total Surface Area (TSA): The sum of the areas of the two circular bases and the curved surface.
$TSA of cylinder = 2 π r (r + h)$
Volume:
Volume of a cylinder is the amount of space enclosed by the cylinder.
$Volume of cylinder = π r^{2} h$

4. Right Circular Cone

A cone has a circular base and a curved surface that tapers to a point called the vertex.
Surface Area:
Curved Surface Area (CSA): The area of the slant or curved surface of the cone.
$CSA of cone = π r l$
Where $r$ is the radius of the base and $l$ is the slant height of the cone.
Total Surface Area (TSA): The sum of the areas of the circular base and the curved surface.
$TSA of cone = π r (l + r)$
The slant height $l$ is related to the radius $r$ and height $h$ by the Pythagorean theorem:
$l = \sqrt{r^{2} + h^{2}}$
Volume:
Volume of a cone is the amount of space enclosed by the cone.
$Volume of cone = \frac{1}{3} π r^{2} h$

5. Sphere

A sphere is a perfectly symmetrical 3D object where every point on the surface is equidistant from the center.
Surface Area:
Surface Area of a sphere:
$Surface area of sphere = 4 π r^{2}$
Where $r$ is the radius of the sphere.
Volume:
Volume of a sphere:
$Volume of sphere = \frac{4}{3} π r^{3}$

6. Hemisphere

A hemisphere is half of a sphere, with one flat circular face and one curved surface.
Surface Area:
Curved Surface Area (CSA) of a hemisphere:
$CSA of hemisphere = 2 π r^{2}$
Total Surface Area (TSA) of a hemisphere (including the base):
$TSA of hemisphere = 3 π r^{2}$
Volume:
Volume of a hemisphere:
$Volume of hemisphere = \frac{2}{3} π r^{3}$

7. Combination of Solids

In practical situations, we may need to find the surface area or volume of objects made by combining two or more of the basic solids. For example:
Frustum of a cone: A cone with the top cut off, forming a smaller circular base at the top. The formulas for frustums can be derived using the concept of similar triangles and the original cone’s properties.
Composite Solids: For objects composed of more than one solid shape (e.g., a cylinder with a hemisphere on top), the surface areas and volumes of the individual shapes are calculated and then added together to find the total surface area or volume.

Summary:

The formulas for surface areas and volumes of cuboids, cubes, cylinders, cones, spheres, and hemispheres are essential for solving real-life problems.
For each solid, there are formulas for both total surface area (including all faces) and volume (the space enclosed by the object).
Understanding how to apply these formulas allows us to calculate dimensions and measurements for complex shapes and structures.