3D Geometry Formulas for Class 11
In Class 11, 3D geometry is an essential topic in mathematics that deals with the study of three-dimensional shapes and figures. It involves the use of various formulas to calculate the surface area, volume, and other properties of these shapes. In this article, we will discuss some of the most important 3D geometry formulas for Class 11 students.
Formulas for 3D Shapes
1. Cube
- Surface Area: 6a², where 'a' is the side length of the cube
- Volume: a³, where 'a' is the side length of the cube
2. Cuboid
- Surface Area: 2(lb + bh + hl), where 'l', 'b', and 'h' are the length, breadth, and height of the cuboid, respectively
- Volume: lbh, where 'l', 'b', and 'h' are the length, breadth, and height of the cuboid, respectively
3. Cylinder
- Surface Area: 2πr(h + r), where 'r' is the radius of the base and 'h' is the height of the cylinder
- Volume: πr²h, where 'r' is the radius of the base and 'h' is the height of the cylinder
4. Cone
- Surface Area: πr(r + l), where 'r' is the radius of the base and 'l' is the slant height of the cone
- Volume: (1/3)πr²h, where 'r' is the radius of the base and 'h' is the height of the cone
5. Sphere
- Surface Area: 4πr², where 'r' is the radius of the sphere
- Volume: (4/3)πr³, where 'r' is the radius of the sphere
Distance and Section Formulas
1. Distance Formula
The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space is given by:
√[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
2. Section Formula
The coordinates of a point that divides the line segment joining two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio m:n are given by:
((mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n), (mz₂ + nz₁) / (m + n))
Important Concepts and Formulas
1. Direction Cosines
The direction cosines of a line in 3D space are given by:
l = cosα, m = cosβ, n = cosγ
where α, β, and γ are the angles that the line makes with the x, y, and z axes, respectively.
2. Angle between Two Lines
The angle between two lines having direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by:
cosθ = |l₁l₂ + m₁m₂ + n₁n₂|
where θ is the angle between the two lines.
Conclusion
These are some of the most important 3D geometry formulas for Class 11 students. Mastering these formulas and concepts is crucial for solving problems in 3D geometry and for scoring well in mathematics exams.
