Resources · Geometry Reference
Geometry Formulas
Geometry formulas are essential tools for measuring physical spaces, building structures, solving math assignments, and understanding the physical world. This comprehensive reference contains a complete list of 2D and 3D formulas for area, perimeter, volume, surface area, coordinate geometry, and right triangles.
12 min read
1. Introduction to Geometry Formulas
Geometry is the branch of mathematics that studies the sizes, shapes, positions, angles, and dimensions of things. Whether analyzing two-dimensional flat polygons or three-dimensional solids, standard formulas let you calculate critical measurements quickly.
To get the most out of this reference, ensure all your input measurements use the same unit system (e.g., all inches or all meters) before calculating. You can browse specific solvers like the area calculator, perimeter calculator, or volume calculator to check your work instantly.
2. Two-Dimensional Area Formulas
Area measures the 2D space enclosed inside a boundary. It is always expressed in square units (e.g., square meters, square feet, or square inches). Read our detailed guide on the Area Formula to study derivations.
| Shape | Formula | Variables & Example |
|---|---|---|
| Square | A = s² | s = side. Example: s = 4m → A = 16 m² |
| Rectangle | A = l × w | l = length, w = width. Example: 5m × 3m → A = 15 m² |
| Triangle | A = ½ × b × h | b = base, h = height. Example: b = 6m, h = 4m → A = 12 m² |
| Circle | A = π × r² | r = radius. Example: r = 3m → A ≈ 28.27 m² |
| Trapezoid | A = ½ × (a + b) × h | a, b = bases, h = height. Example: a = 4, b = 6, h = 3 → A = 15 m² |
| Parallelogram | A = b × h | b = base, h = height. Example: b = 8m, h = 4m → A = 32 m² |
| Rhombus | A = ½ × d₁ × d₂ | d₁, d₂ = diagonals. Example: d₁ = 6, d₂ = 8 → A = 24 m² |
Need to solve a specific shape? Use our specialty solvers: Rectangle Calculator, Triangle Area Calculator, Trapezoid Calculator, or Parallelogram Calculator.
3. Perimeter and Circumference Formulas
Perimeter is the 1-dimensional boundary length around the outside edge of a polygon. For circles, the boundary distance is called the circumference. These metrics are expressed in linear units (like meters or feet). See our guide on the Perimeter Formula and the Circumference Formula for step-by-step math.
| Shape | Formula | Variables & Example |
|---|---|---|
| Square | P = 4s | s = side. Example: s = 4m → P = 16 m |
| Rectangle | P = 2(l + w) | l = length, w = width. Example: l = 5m, w = 3m → P = 16 m |
| Triangle | P = a + b + c | a, b, c = side lengths. Example: sides 3m, 4m, 5m → P = 12 m |
| Circle (Circumference) | C = 2πr | r = radius. Example: r = 3m → C ≈ 18.85 m |
Run fast boundary calculations using the Circumference Calculator or the comprehensive Regular Polygon Calculator.
4. Three-Dimensional Volume Formulas
Volume measures the amount of three-dimensional space enclosed inside a solid object. It is expressed in cubic units (e.g., cubic meters, cubic feet, or cubic inches). Check out the detailed Volume Formula guide.
| Solid Shape | Volume Formula | Variables & Example |
|---|---|---|
| Cube | V = s³ | s = side length. Example: s = 3m → V = 27 m³ |
| Rectangular Prism | V = l × w × h | l = length, w = width, h = height. Example: 4 × 3 × 5 → V = 60 m³ |
| Cylinder | V = π × r² × h | r = radius, h = height. Example: r = 2m, h = 5m → V ≈ 62.83 m³ |
| Sphere | V = 4/3 × π × r³ | r = radius. Example: r = 3m → V ≈ 113.10 m³ |
| Cone | V = 1/3 × π × r² × h | r = radius, h = height. Example: r = 3m, h = 4m → V ≈ 37.70 m³ |
| Pyramid | V = 1/3 × B × h | B = base area, h = height. Example: B = 9 m², h = 4m → V = 12 m³ |
To compute capacities instantly, open our Volume Calculator or the dedicated Cone Volume Calculator.
5. Surface Area Formulas
Surface area is the total area of all the outer faces of a 3D solid. It represents a 2-dimensional boundary size and is expressed in square units. Study the underlying math in our Surface Area Formula article.
| Solid Shape | Surface Area Formula | Variables & Example |
|---|---|---|
| Cube | SA = 6s² | s = side length. Example: s = 3m → SA = 54 m² |
| Rectangular Prism | SA = 2(lw + lh + wh) | l = len, w = width, h = height. Example: 4 × 3 × 5 → SA = 94 m² |
| Cylinder | SA = 2πr(r + h) | r = radius, h = height. Example: r = 2m, h = 5m → SA ≈ 87.96 m² |
| Sphere | SA = 4πr² | r = radius. Example: r = 3m → SA ≈ 113.10 m² |
| Cone | SA = πr(r + l) | r = radius, l = slant height. Example: r = 3, l = 5 → SA ≈ 75.40 m² |
Perform fast calculations on 3D faces using the Surface Area Calculator.
6. Coordinate Geometry Formulas
Coordinate geometry uses algebraic formulas to analyze line segments and shapes plotted on a Cartesian coordinate plane.
| Concept | Formula | Example Calculation |
|---|---|---|
| Distance Formula | d = √((x₂ − x₁)² + (y₂ − y₁)*2) | From (1, 2) to (4, 6) → d = √(3² + 4²) = 5 |
| Midpoint Formula | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) | Between (1, 2) and (5, 8) → M = (3, 5) |
| Slope Formula | m = (y₂ − y₁) / (x₂ − x₁) | From (1, 2) to (3, 6) → m = (6 − 2)/(3 − 1) = 2 |
Verify lines and coordinates instantly with these grid tools: Distance Formula Calculator, Midpoint Formula Calculator, and Slope Calculator.
7. Right Triangle Formulas and Ratios
Right-angled triangles have special mathematical properties. The Pythagorean theorem links side lengths, while basic trigonometric ratios link angles to side ratios.
Pythagorean Theorem
a² + b² = c²
Where c represents the hypotenuse (the longest side opposite the right angle), and a and b represent the other two perpendicular sides. Example: a = 3, b = 4 → c = √(9 + 16) = 5.
Trigonometric Ratios (SOH CAH TOA)
- sin(θ) = Opposite / Hypotenuse
- cos(θ) = Adjacent / Hypotenuse
- tan(θ) = Opposite / Adjacent
These ratios relate an acute angle θ in a right triangle to ratio lengths of its sides.
Perform right-triangle math using the Pythagorean Theorem Calculator, Right Triangle Calculator, or explore trigonometry values with the Unit Circle Calculator.
8. Circle Geometry Formulas
Circles have unique geometric equations because of their curved boundaries. These formulas rely on the radius (r, distance from center to edge) and the constant pi (π ≈ 3.14159).
| Metric | Equation | Variables & Example |
|---|---|---|
| Area | A = π × r² | r = radius. Example: r = 5m → A ≈ 78.54 m² |
| Circumference | C = 2 × π × r | r = radius. Example: r = 5m → C ≈ 31.42 m |
| Arc Length | L = (θ / 360°) × 2πr | θ = angle in degrees. Example: r = 6, θ = 60° → L ≈ 6.28 m |
| Sector Area | A_sector = (θ / 360°) × πr² | θ = angle in degrees. Example: r = 6, θ = 60° → A ≈ 18.85 m² |
To work with sectors and arcs, see the Arc Length Calculator, read the resource article on the Arc Length Formula, or use the general Circle Calculator to solve for diameter and radius.
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Frequently asked questions
The most common geometry formulas are for finding area (e.g., A = l × w for rectangles, A = πr² for circles), perimeter (e.g., P = 2l + 2w), and volume (e.g., V = lwh for boxes, V = πr²h for cylinders).
Perimeter measures the 1-dimensional boundary length around the outside of a 2D shape (measured in linear units like feet). Area measures the 2-dimensional flat space inside the boundary (measured in square units like square feet).
For prisms and cylinders, volume is base area multiplied by height (V = Base × h). For cones and pyramids, it is 1/3 of the base area times height (V = 1/3 × Base × h). For spheres, it is 4/3 × π × r³.
Volume measures the total 3-dimensional space contained inside a shape (measured in cubic units). Surface area measures the total 2-dimensional area of all the outer surfaces of the shape combined (measured in square units).
The Pythagorean theorem states that in a right-angled triangle, the square of the longest side (hypotenuse, c) is equal to the sum of the squares of the other two shorter sides (a and b): a² + b² = c².
Coordinate geometry formulas—such as distance, midpoint, and slope—are used to analyze lines, points, and segments plotted on a 2D Cartesian grid using coordinate points (x, y).