Surface Areas and Volumes Class 10 Mathematics Concept notes and Formula Sheet

 

Concepts:

1. Introduction to Surface Areas and Volumes:

   • Surface Area refers to the total area covered by the surface of a three-dimensional object.
   • Volume refers to the amount of space enclosed within a three-dimensional object.

2. Cuboid:

   • A cuboid is a 3D shape with six rectangular faces.
   • Surface Area:
     • Lateral Surface Area (LSA): The sum of the areas of all the lateral (side) faces. $$\text{LSA} = 2h(l + b)$$
     • Total Surface Area (TSA): The sum of the areas of all six faces. $$\text{TSA} = 2(lb + bh + hl)$$
     • Volume: $$V = l \times b \times h$$
     where $l$, $b$, and $h$ are the length, breadth, and height of the cuboid.

3. Cube:

   • A cube is a special case of a cuboid where all sides are equal.
   • Surface Area:
     • Lateral Surface Area: $$\text{LSA} = 4a^2$$
     • Total Surface Area: $$\text{TSA} = 6a^2$$
     • Volume: $$V = a^3$$
     where $a$ is the side of the cube.

4. Right Circular Cylinder:

   • A cylinder is a 3D shape with two parallel circular bases connected by a curved surface.
   • Surface Area:
     • Curved Surface Area (CSA): $$\text{CSA} = 2\pi rh$$
     • Total Surface Area: $$\text{TSA} = 2\pi r(r + h)$$
     • Volume: $$V = \pi r^2 h$$
     where $r$ is the radius of the base and $h$ is the height of the cylinder.

5. Right Circular Cone:

   • A cone is a 3D shape with a circular base and a single vertex.
   • Surface Area:
     • Curved Surface Area (CSA): $$\text{CSA} = \pi rl$$
     • Total Surface Area: $$\text{TSA} = \pi r(r + l)$$
     • Volume: $$V = \frac{1}{3}\pi r^2 h$$
     where $r$ is the radius of the base, $h$ is the height, and $l$ is the slant height of the cone. $$l = \sqrt{r^2 + h^2}$$

6. Sphere:

   • A sphere is a perfectly round 3D shape where all points on the surface are equidistant from the center.
   • Surface Area: $$\text{SA} = 4\pi r^2$$
   • Volume: $$V = \frac{4}{3}\pi r^3$$ where $r$ is the radius of the sphere.

7. Hemisphere:

   • A hemisphere is half of a sphere.
   • Surface Area:
     • Curved Surface Area: $$\text{CSA} = 2\pi r^2$$
     • Total Surface Area: $$\text{TSA} = 3\pi r^2$$
     • Volume: $$V = \frac{2}{3}\pi r^3$$
     where $r$ is the radius of the hemisphere.

8. Frustum of a Cone:

   • A frustum is the portion of a cone obtained by cutting it parallel to the base.
   • Surface Area:
     • Curved Surface Area: $$\text{CSA} = \pi (r_1 + r_2)l$$
     • Total Surface Area: $$\text{TSA} = \pi \left[r_1^2 + r_2^2 + (r_1 + r_2)l \right]$$
     • Volume: $$V = \frac{1}{3} \pi h \left( r_1^2 + r_2^2 + r_1r_2 \right)$$
     where $r_1$ and $r_2$ are the radii of the two parallel circular faces, $h$ is the height, and $l$ is the slant height of the frustum: $$l = \sqrt{h^2 + (r_1 - r_2)^2}$$

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Formula Sheet:

1. Cuboid:

   • LSA: $2h(l + b)$
   • TSA: $2(lb + bh + hl)$
   • Volume: $l \times b \times h$

2. Cube:

   • LSA: $4a^2$
   • TSA: $6a^2$
   • Volume: $a^3$

3. Cylinder:

   • CSA: $2\pi rh$
   • TSA: $2\pi r(r + h)$
   • Volume: $\pi r^2 h$

4. Cone:

   • CSA: $\pi rl$
   • TSA: $\pi r(r + l)$
   • Volume: $\frac{1}{3}\pi r^2 h$

5. Sphere:

   • SA: $4\pi r^2$
   • Volume: $\frac{4}{3}\pi r^3$

6. Hemisphere:

   • CSA: $2\pi r^2$
   • TSA: $3\pi r^2$
   • Volume: $\frac{2}{3}\pi r^3$

7. Frustum of a Cone:

   • CSA: $\pi (r_1 + r_2)l$
   • TSA: $\pi \left[r_1^2 + r_2^2 + (r_1 + r_2)l \right]$
   • Volume: $\frac{1}{3} \pi h \left( r_1^2 + r_2^2 + r_1r_2 \right)$