surface areas and volumes class 10 formulas
Surface Areas and Volumes: Class 10 Formulas
Let’s delve into the essential formulas related to surface areas and volumes from the Class 10 curriculum. These concepts involve understanding the surface area and volume of different 3-dimensional shapes such as cubes, cuboids, cylinders, cones, spheres, and hemispheres. Here, I will provide detailed explanations, derivations, and examples for each of the shapes.
1. Cuboid
A cuboid is a 3-dimensional shape with six rectangular faces.
- Surface Area: The total area covered by the surface of a cuboid.
- Formula: 2(lb + bh + hl)
- Where ( l ) is the length, ( b ) is the breadth, and ( h ) is the height of the cuboid.
- Volume: The amount of space occupied by the cuboid.
- Formula: l \times b \times h
- Lateral Surface Area (LSA): The sum of the areas of all the side faces (excluding top and bottom).
- Formula: 2h(l + b)
2. Cube
A cube is a special type of cuboid where all sides are equal.
- Surface Area: All six faces are squares.
- Formula: 6a^2
- Where ( a ) is the side of the cube.
- Volume: Space occupied by the cube.
- Formula: a^3
- Lateral Surface Area: The area excluding the top and bottom faces.
- Formula: 4a^2
3. Cylinder
A cylinder has two parallel circular bases connected by a curved surface.
- Curved Surface Area (CSA): Area of the curved surface.
- Formula: 2\pi rh
- Where ( r ) is the radius and ( h ) is the height.
- Total Surface Area (TSA): Sum of curved surface area and the area of the two bases.
- Formula: 2\pi r(h + r)
- Volume: Space occupied by the cylinder.
- Formula: \pi r^2 h
4. Cone
A cone has a circular base and a pointed top.
- Curved Surface Area: Area of the curved surface.
- Formula: \pi rl
- Where ( l ) (slant height) can be found using ( l = \sqrt{r^2 + h^2} ).
- Total Surface Area: Sum of curved surface area and base area.
- Formula: \pi r(l + r)
- Volume: Space occupied by the cone.
- Formula: \frac{1}{3}\pi r^2 h
5. Sphere
A perfect 3D circular shape.
- Surface Area: Only one spherical surface.
- Formula: 4\pi r^2
- Where ( r ) is the radius.
- Volume: Space occupied by the sphere.
- Formula: \frac{4}{3}\pi r^3
6. Hemisphere
A hemisphere is half of a sphere.
- Curved Surface Area: The curved surface only.
- Formula: 2\pi r^2
- Total Surface Area: Curved surface + base area.
- Formula: 3\pi r^2
- Volume: Space occupied by the hemisphere.
- Formula: \frac{2}{3}\pi r^3
Detailed Examples and Applications
To illustrate how these formulas are applied, let’s work through an example for each shape.
Example for Cuboid
Problem: Find the total surface area and volume of a cuboid with length 5 cm, breadth 4 cm, and height 3 cm.
- Solution:
- Surface Area = 2(5 \times 4 + 4 \times 3 + 5 \times 3) = 2(20 + 12 + 15) = 94 \, \text{cm}^2.
- Volume = 5 \times 4 \times 3 = 60 \, \text{cm}^3.
Example for Cube
Problem: Calculate the total surface area and volume of a cube with side 6 cm.
- Solution:
- Surface Area = 6 \times 6^2 = 6 \times 36 = 216 \, \text{cm}^2.
- Volume = 6^3 = 216 \, \text{cm}^3.
Example for Cylinder
Problem: Determine the curved and total surface area, and volume of a cylinder with radius 2 cm and height 7 cm.
- Solution:
- Curved Surface Area = 2\pi \times 2 \times 7 = 28\pi \, \text{cm}^2.
- Total Surface Area = 2\pi \times 2 \times (7 + 2) = 36\pi \, \text{cm}^2.
- Volume = \pi \times 2^2 \times 7 = 28\pi \, \text{cm}^3.
Example for Cone
Problem: Find the total surface area and volume of a cone with radius 3 cm and height 4 cm.
- Solution:
- Slant height ( l = \sqrt{3^2 + 4^2} = 5 , \text{cm}).
- Curved Surface Area = \pi \times 3 \times 5 = 15\pi \, \text{cm}^2.
- Total Surface Area = \pi \times 3 \times (3 + 5) = 24\pi \, \text{cm}^2.
- Volume = \frac{1}{3}\pi \times 3^2 \times 4 = 12\pi \, \text{cm}^3.
Example for Sphere
Problem: Find the surface area and volume of a sphere with radius 4 cm.
- Solution:
- Surface Area = 4\pi \times 4^2 = 64\pi \, \text{cm}^2.
- Volume = \frac{4}{3}\pi \times 4^3 = \frac{256}{3}\pi \text{cm}^3.
Example for Hemisphere
Problem: Calculate the curved surface area, total surface area, and volume of a hemisphere with radius 5 cm.
- Solution:
- Curved Surface Area = 2\pi \times 5^2 = 50\pi \, \text{cm}^2.
- Total Surface Area = 3\pi \times 5^2 = 75\pi \, \text{cm}^2.
- Volume = \frac{2}{3}\pi \times 5^3 = \frac{250}{3}\pi \, \text{cm}^3.
Understanding and applying these formulas can be extremely useful in solving practical problems related to real-life objects that can be approximated by these geometric shapes. Always remember to substitute the given measurements into the correct formula, perform the arithmetic accurately, and include the proper units in your final answers. If you have any further questions or need more clarification on any of these shapes, feel free to ask!