3D Geometry Formulas
2026-02-28 10:30 Diff
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Last updated on August 10, 2025

In 3D geometry, we explore various formulas related to solids, their volumes, surface areas, and coordinates in three-dimensional space. This topic covers the essential formulas needed for understanding 3D geometry in mathematics.

List of 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.

Volume Formulas

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 ).

Surface Area Formulas

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.

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Coordinate Geometry in 3D

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 ).

Importance of 3D Geometry Formulas

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.

Tips and Tricks to Memorize 3D Geometry Formulas

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.

Common Mistakes and How to Avoid Them While Using 3D Geometry Formulas

Students often make errors when applying 3D geometry formulas. Here are some common mistakes and tips to avoid them.

Problem 1

Find the volume of a cube with side length 4 cm.

Okay, lets begin

The volume is 64 cm³

Explanation

To find the volume, use the formula ( V = a3 ).

Given side length ( a = 4 ) cm,

Volume ( V = 43 = 64 ) cm³.

Well explained 👍

Problem 2

Calculate the surface area of a sphere with a radius of 5 cm.

Okay, lets begin

The surface area is 314 cm²

Explanation

To find the surface area, use the formula ( SA = 4 pi r2 ).

Given radius ( r = 5 ) cm,

Surface area ( SA = 4 pi (5)2 = 314 ) cm² (approximating (pi) as 3.14).

Well explained 👍

Problem 3

Find the distance between points (1, 2, 3) and (4, 6, 8) in 3D space.

Okay, lets begin

The distance is 7 units

Explanation

To find the distance, use the formula ( d = sqrt{(x_2-x_1)2 + (y_2-y_1)2 + (z_2-z_1)2} ).

Substitute the given coordinates: ( d = sqrt{(4-1)2 + (6-2)2 + (8-3)2} = sqrt{32 + 42 + 52} = sqrt{9 + 16 + 25} = sqrt{50} approx 7 ) units.

Well explained 👍

Problem 4

Find the volume of a cylinder with radius 3 cm and height 7 cm.

Okay, lets begin

The volume is 198 cm³

Explanation

To find the volume, use the formula ( V = pi r2 h ).

Given radius ( r = 3 ) cm and height ( h = 7 ) cm,

Volume ( V = pi (3)2 (7) = 198 ) cm³ (approximating (pi) as 3.14).

Well explained 👍

Problem 5

Calculate the surface area of a cuboid with dimensions 2 cm by 3 cm by 4 cm.

Okay, lets begin

The surface area is 52 cm²

Explanation

To find the surface area, use the formula ( SA = 2(lb + bh + hl) ).

Given ( l = 2 ) cm, ( b = 3 ) cm, ( h = 4 ) cm,

Surface area ( SA = 2(2 times 3 + 3 times 4 + 4 times 2) = 2(6 + 12 + 8) = 52 ) cm².

Well explained 👍

FAQs on 3D Geometry Formulas

1.What is the volume formula for a sphere?

The formula to find the volume of a sphere is ( V = frac{4}{3} pi r3 ).

2.How do you find the surface area of a cone?

The formula for the surface area of a cone is ( SA = pi r(l + r) ), where ( l ) is the slant height.

3.What is the distance formula in 3D geometry?

The distance formula between two points ((x_1, y_1, z_1)) and ((x_2, y_2, z_2)) is ( d = sqrt{(x_2-x_1)2 + (y_2-y_1)2 + (z_2-z_1)2}z).

4.How do you find the volume of a cone?

The volume of a cone is found using the formula ( V = frac{1}{3} pi r2 h ).

5.What is the section formula in 3D geometry?

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

Glossary for 3D Geometry Formulas

  • 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 ).

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.