3D Geometry Formulas

In 3D geometry, key formulas include those for calculating volumes, surface areas, and coordinates of points in space. Let’s learn these essential formulas for 3D geometry.

The volume of a solid is the amount of space it occupies.

Here are some key volume formulas: 

• Volume of a cube: ( V = a³ ), where ( a )is the side length.
   
• Volume of a cuboid: ( V = l ×b times h ), where ( l ), ( b ), and ( h ) are the length, breadth, and height. 
   
• Volume of a sphere: ( V = frac{4}{3} pi r³ ), where ( r ) is the radius. 
   
• Volume of a cylinder: ( V = pi r² h ), where ( r ) is the radius, and ( h ) is the height. 
   
• Volume of a cone: ( V = frac{1}{3} pi r² h ).

The surface area of a solid is the total area of its outer surfaces. Key surface area formulas include:

• Surface area of a cube: ( SA = 6a² ). 
   
• Surface area of a cuboid: ( SA = 2(lb + bh + hl) ).
   
• Surface area of a sphere: ( SA = 4 pi r² ). 
   
• Surface area of a cylinder: ( SA = 2 pi r(h + r) ). 
   
• Surface area of a cone: ( SA = pi r(l + r) ), where ( l ) is the slant height.

In 3D coordinate geometry, we use coordinates to locate points in space. Key formulas include: -

Distance between two points ((x_1, y_1, z_1)) and ((x_2, y_2, z_2)): ( d = sqrt{(x_2-x_1)² + (y_2-y_1)² + (z_2-z_1)²} ). -

Section formula for a point dividing a line segment in the ratio ( m:n ): ((frac{mx_2+nx_1}{m+n}, frac{my_2+ny_1}{m+n}, frac{mz_2+nz_1}{m+n})). -

Equation of a plane in normal form: ( ax + by + cz = d ).

In mathematics and real-world applications, 3D geometry formulas are crucial for analyzing and understanding spatial structures.

Understanding these formulas aids in: 

• Calculating the volumes and surface areas of various solids in engineering and architecture. 
   
• Solving problems in physics related to motion and forces in three dimensions. 
   
• Modeling and interpreting data in fields like computer graphics and virtual reality.

Many students find 3D geometry formulas challenging.

Here are some tips to master them: 

• Visualize each shape and relate it to real-world objects to understand its dimensions better. 
   
• Use mnemonic devices to remember the order of terms in formulas, such as "Pi r squared height" for the volume of a cylinder. 
   
• Create flashcards with diagrams and formulas to reinforce memory through repetition.

• Volume: The amount of space occupied by a solid. Calculated using formulas specific to each shape.

• Surface Area: The total area of the external surfaces of a solid. Different shapes have specific formulas.

• 3D Coordinates: A system to specify the position of a point in space using three values (x, y, z).

• Slant Height: The diagonal height along the side of a cone or pyramid, used in surface area calculations.

• Plane Equation: An equation representing a flat surface in 3D space, typically expressed as ( ax + by + cz = d ).