All Geometric Formulas
Complete Reference Guide for 2D & 3D Shapes
Introduction to Geometric Formulas
Geometric formulas are mathematical equations used to calculate various properties of shapes and figures, including area, perimeter, volume, and surface area[web:101][web:102]. These formulas are fundamental tools in mathematics, engineering, architecture, and numerous real-world applications[web:110][web:113].
Geometry is divided into two main categories[web:106]:
- 2D Geometry (Plane Geometry): Deals with flat shapes like circles, triangles, rectangles, and squares[web:102][web:106]
- 3D Geometry (Solid Geometry): Concerns three-dimensional objects like cubes, spheres, cylinders, and cones[web:102][web:103]
This comprehensive guide covers all essential geometric formulas needed for academic studies and practical applications across various curricula including IB, AP, GCSE, IGCSE, and more[web:101][web:114].
2D Geometry Formulas
Two-dimensional shapes lie flat on a plane and have only length and width[web:101][web:102].
1. Square
A square has four equal sides and four right angles[web:102][web:106].
Where \( a \) = side length
• Area: \( A = a^2 \)
• Perimeter: \( P = 4a \)
• Diagonal: \( d = a\sqrt{2} \)
2. Rectangle
A rectangle has opposite sides equal and four right angles[web:101][web:102].
Where \( l \) = length, \( w \) = width
• Area: \( A = l \times w \)
• Perimeter: \( P = 2(l + w) \)
• Diagonal: \( d = \sqrt{l^2 + w^2} \)
3. Triangle
A three-sided polygon with three angles[web:101][web:102].
Where \( b \) = base, \( h \) = height, \( a, b, c \) = sides
• Area: \( A = \frac{1}{2} \times b \times h \)
• Perimeter: \( P = a + b + c \)
• Heron's Formula: \( A = \sqrt{s(s-a)(s-b)(s-c)} \) where \( s = \frac{a+b+c}{2} \)
• Pythagorean Theorem (right triangle): \( a^2 + b^2 = c^2 \)
4. Circle
A perfectly round shape where all points are equidistant from the center[web:102][web:106].
Where \( r \) = radius, \( d \) = diameter
• Area: \( A = \pi r^2 \)
• Circumference: \( C = 2\pi r = \pi d \)
• Area of Sector: \( A = \frac{\theta}{360°} \times \pi r^2 \)
• Arc Length: \( L = \frac{\theta}{360°} \times 2\pi r \)
5. Parallelogram
A quadrilateral with opposite sides parallel and equal[web:102][web:106].
Where \( b \) = base, \( h \) = height, \( a \) = side
• Area: \( A = b \times h \)
• Perimeter: \( P = 2(a + b) \)
6. Rhombus
A quadrilateral with all four sides equal[web:102].
Where \( d_1, d_2 \) = diagonals, \( a \) = side
• Area: \( A = \frac{1}{2} \times d_1 \times d_2 \)
• Perimeter: \( P = 4a \)
7. Trapezoid (Trapezium)
A quadrilateral with one pair of parallel sides[web:102].
Where \( a, b \) = parallel sides, \( h \) = height
• Area: \( A = \frac{1}{2}(a + b) \times h \)
• Perimeter: \( P = \) sum of all sides
3D Geometry Formulas
Three-dimensional shapes have length, width, and height (depth)[web:102][web:103][web:105].
1. Cube
A solid with six square faces of equal size[web:103][web:105].
Where \( a \) = edge length
• Volume: \( V = a^3 \)
• Total Surface Area: \( TSA = 6a^2 \)
• Lateral Surface Area: \( LSA = 4a^2 \)
• Diagonal: \( d = a\sqrt{3} \)
2. Cuboid (Rectangular Prism)
A solid with six rectangular faces[web:103][web:105][web:106].
Where \( l \) = length, \( w \) = width, \( h \) = height
• Volume: \( V = l \times w \times h \)
• Total Surface Area: \( TSA = 2(lw + wh + lh) \)
• Lateral Surface Area: \( LSA = 2h(l + w) \)
• Diagonal: \( d = \sqrt{l^2 + w^2 + h^2} \)
3. Sphere
A perfectly round three-dimensional object[web:103][web:105].
Where \( r \) = radius
• Volume: \( V = \frac{4}{3}\pi r^3 \)
• Surface Area: \( SA = 4\pi r^2 \)
4. Cylinder
A solid with two parallel circular bases[web:103][web:105].
Where \( r \) = radius, \( h \) = height
• Volume: \( V = \pi r^2 h \)
• Total Surface Area: \( TSA = 2\pi r(r + h) \)
• Curved Surface Area: \( CSA = 2\pi rh \)
5. Cone
A solid with a circular base tapering to a point[web:103][web:105][web:112].
Where \( r \) = radius, \( h \) = height, \( l \) = slant height
• Volume: \( V = \frac{1}{3}\pi r^2 h \)
• Total Surface Area: \( TSA = \pi r(r + l) \)
• Curved Surface Area: \( CSA = \pi rl \)
• Slant Height: \( l = \sqrt{r^2 + h^2} \)
6. Pyramid
A solid with a polygonal base and triangular faces meeting at an apex[web:105].
Where \( B \) = base area, \( h \) = height
• Volume: \( V = \frac{1}{3}Bh \)
• Surface Area: Base area + Sum of triangular faces
7. Prism
A solid with two identical polygonal bases and rectangular sides[web:105].
Where \( B \) = base area, \( h \) = height, \( P \) = base perimeter
• Volume: \( V = Bh \)
• Lateral Surface Area: \( LSA = Ph \)
Quick Reference Table
Summary of common geometric formulas for quick access[web:101][web:114]:
| Shape | Area/Volume | Perimeter/Surface Area |
|---|---|---|
| Square | \( a^2 \) | \( 4a \) |
| Rectangle | \( l \times w \) | \( 2(l + w) \) |
| Triangle | \( \frac{1}{2}bh \) | \( a + b + c \) |
| Circle | \( \pi r^2 \) | \( 2\pi r \) |
| Cube | \( a^3 \) | \( 6a^2 \) |
| Cuboid | \( lwh \) | \( 2(lw + wh + lh) \) |
| Sphere | \( \frac{4}{3}\pi r^3 \) | \( 4\pi r^2 \) |
| Cylinder | \( \pi r^2 h \) | \( 2\pi r(r + h) \) |
| Cone | \( \frac{1}{3}\pi r^2 h \) | \( \pi r(r + l) \) |
Real-World Applications
Geometric formulas are essential in countless real-world scenarios[web:110][web:113][web:118]:
🏗️ Architecture & Construction
Calculating building materials, floor space, wall areas, and structural design[web:110][web:113].
🎨 Interior Design
Measuring spaces for furniture, flooring, paint, and decorative elements[web:118].
🍳 Cooking & Baking
Measuring ingredients, calculating pan sizes, and portion control[web:110][web:118].
🌳 Gardening & Landscaping
Planning garden beds, calculating fertilizer needs, and designing outdoor spaces[web:118].
📦 Packaging & Shipping
Designing efficient packaging, calculating shipping volumes, and optimizing storage[web:110].
🎮 Computer Graphics
Creating 3D models, rendering scenes, and developing video games[web:110].
🌾 Agriculture
Measuring field areas, irrigation planning, and crop yield estimation.
⚙️ Engineering
Mechanical design, structural analysis, and manufacturing processes[web:113].
🏥 Medical Imaging
CT scans, MRI analysis, and calculating medication dosages based on body surface area.
🎯 Sports
Designing playing fields, calculating court dimensions, and analyzing trajectories.
Important Facts & Tips
💡 Unit Consistency
Always ensure all measurements are in the same units before applying formulas. Area uses square units, volume uses cubic units.
💡 Pi Value
Use \( \pi \approx 3.14159 \) or \( \frac{22}{7} \) for calculations. Most calculators have a π button for precision.
💡 Surface Area vs Volume
Surface area measures the outside covering of a 3D shape (square units), while volume measures the space inside (cubic units).
💡 Pythagorean Theorem
This fundamental theorem \( a^2 + b^2 = c^2 \) is essential for many geometric calculations and appears in multiple formulas[web:101][web:113].
💡 Composite Shapes
For complex shapes, break them into simpler components, calculate each separately, then add or subtract as needed.
💡 Ancient Origins
Ancient Egyptians and Babylonians developed many geometric formulas over 4000 years ago for construction and land measurement.
💡 Formula Derivations
Understanding how formulas are derived helps with memorization and application. Many formulas are related to each other.
💡 Curriculum Coverage
These formulas appear across all mathematics curricula including IB Math, AP Geometry, GCSE/IGCSE, A-Level, SAT, ACT, and international programs.
Common Mistakes to Avoid
❌ Mixing Units
Don't mix centimeters with meters or inches with feet in the same calculation. Convert all to one unit first.
❌ Forgetting Squared/Cubed
Remember: area is always in square units (\( m^2 \)), volume in cubic units (\( m^3 \)).
❌ Wrong Formula Selection
Using the wrong formula for a shape (e.g., circle formula for an ellipse) leads to incorrect results.
❌ Diameter vs Radius
Remember: diameter = 2 × radius. Using diameter in a radius formula multiplies your error by 4!
❌ Order of Operations
Follow PEMDAS/BODMAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
Study Tips for Mastering Geometric Formulas
✓ Create Formula Cards
Make flashcards with shape diagrams on one side and formulas on the other for quick revision.
✓ Practice Regularly
Solve problems daily to reinforce formula application and build confidence.
✓ Visualize Shapes
Draw diagrams and label dimensions. Visual learning enhances understanding and memory.
✓ Group Similar Formulas
Study related shapes together (e.g., all quadrilaterals, all circular shapes) to identify patterns.
✓ Real-World Applications
Look for geometry in everyday life - rooms, packaging, nature - to make formulas more meaningful.
👨🏫 About the Author
Adam
Co-Founder @RevisionTown
Math Expert in various curricula including IB, AP, GCSE, IGCSE, and more.