Complete Guide to Distance Formulas: Comprehensive Table with Explanations
Introduction
Distance formulas are fundamental mathematical tools used across various disciplines including geometry, physics, and calculus. This comprehensive guide covers all essential distance formulas that students from classes 9-12 and college levels need to master.
Complete Distance Formulas Reference Table
| Formula Name | Formula | Variables | Application/Context | Class Level |
|---|---|---|---|---|
| Basic Distance Formula (2D) | d = √[(x₂ – x₁)² + (y₂ – y₁)²] | x₁, y₁ = coordinates of first point; x₂, y₂ = coordinates of second point; d = distance | Finding distance between two points in a plane | Class 10 |
| Distance Formula (3D) | d = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²] | x, y, z = coordinates in 3D space; d = distance | Finding distance between two points in three-dimensional space | Class 12 |
| Speed-Distance-Time Formula | Distance = Speed × Time OR d = s × t | d = distance; s = speed; t = time | Calculating distance traveled at constant speed | Class 7-10 |
| Speed Formula | Speed = Distance ÷ Time OR s = d/t | s = speed; d = distance; t = time | Finding rate of motion | Class 7-10 |
| Time Formula | Time = Distance ÷ Speed OR t = d/s | t = time; d = distance; s = speed | Calculating duration of travel | Class 7-10 |
| Distance in Uniformly Accelerated Motion | s = ut + ½at² | s = distance; u = initial velocity; t = time; a = acceleration | Physics: motion with constant acceleration | Class 9-11 |
| Distance using Equations of Motion | v² = u² + 2as OR s = (v² – u²)/2a | v = final velocity; u = initial velocity; a = acceleration; s = distance | Physics: finding displacement with velocity change | Class 9-11 |
| Euclidean Distance Formula | d = √[Σ(xᵢ – yᵢ)²] for n dimensions | xᵢ, yᵢ = coordinates in n-dimensional space | Data science, machine learning, multidimensional geometry | College |
| Manhattan Distance Formula | d = Σ|xᵢ – yᵢ| for n dimensions | xᵢ, yᵢ = coordinates; Σ = sum of absolute differences | Computer science, grid-based pathfinding, taxi-cab geometry | College |
| Perpendicular Distance from Point to Line (2D) | d = |ax₁ + by₁ + c|/√(a² + b²) | Line: ax + by + c = 0; Point: (x₁, y₁) | Finding shortest distance from point to line | Class 11-12 |
| Perpendicular Distance from Point to Plane | d = |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²) | Plane: ax + by + cz + d = 0; Point: (x₁, y₁, z₁) | 3D geometry, analytical geometry | Class 12 |
| Distance Between Two Parallel Lines | d = |c₂ – c₁|/√(a² + b²) | Lines: ax + by + c₁ = 0 and ax + by + c₂ = 0 | Finding separation between parallel lines | Class 11-12 |
| Shortest Distance Between Two Parallel Lines (3D) | d = |(a₂ – a₁) · (b₁ × b₂)|/|b₁ × b₂| | Lines in vector form; a = position vector; b = direction vector; × = cross product | 3D vector geometry | Class 12 |
| Shortest Distance Between Two Skew Lines | d = |(a₂ – a₁) · (b₁ × b₂)|/|b₁ × b₂| | Same as parallel lines formula; applies when lines don’t intersect | Finding minimum distance between non-intersecting 3D lines | Class 12 |
| Distance of Closest Approach | d = |v₁t – v₂t| or using relative velocity concepts | v₁, v₂ = velocities of objects; t = time at closest approach | Physics: minimum distance between moving objects | Class 11-12 |
| Stopping Distance Formula | d = v²/2μg OR d = reaction distance + braking distance | v = initial velocity; μ = friction coefficient; g = gravity (9.8 m/s²) | Vehicle safety, physics of motion | Class 11, Driving Theory |
| Braking Distance | d = v²/2μg | v = velocity; μ = coefficient of friction; g = acceleration due to gravity | Distance traveled while braking | Class 11 |
| Reaction Distance | d = v × t_reaction | v = velocity; t_reaction = driver reaction time (typically 1.5-2 seconds) | Distance traveled during reaction time | Practical Applications |
| Cosine Distance | cos(θ) = (A · B)/( | A | ||
| Hamming Distance | d = count of positions where symbols differ | Compare two strings/sequences of equal length | Computer science, error detection, coding theory | College |
| Distance in Projectile Motion (Horizontal) | R = (u² sin2θ)/g | R = range; u = initial velocity; θ = angle of projection; g = gravity | Maximum horizontal distance in projectile motion | Class 11 |
| Height and Distance Formulas (Trigonometry) | h = d × tan(θ) OR d = h/tan(θ) | h = height; d = horizontal distance; θ = angle of elevation/depression | Surveying, navigation, solving height problems | Class 10 |
| Distance Using Trigonometric Ratios | d = h × cot(θ) OR h = d × tan(θ) | Various arrangements based on known angle and side | Applications in height and distance problems | Class 10 |
| Arc Length (Distance Along Curve) | s = rθ (θ in radians) | s = arc length; r = radius; θ = angle in radians | Distance along circular path | Class 11 |
| Distance in Polar Coordinates | d = √[r₁² + r₂² – 2r₁r₂cos(θ₂ – θ₁)] | r₁, r₂ = radial distances; θ₁, θ₂ = angles | Converting polar to distance measurement | Class 12, College |
Detailed Explanations of Main Formulas
1. Basic Distance Formula (Coordinate Geometry)
The distance formula calculates the straight-line distance between two points in a coordinate plane. Derived from the Pythagorean theorem, it treats the horizontal and vertical differences as the two sides of a right triangle.
Formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²]
Example: Find distance between points A(3, 4) and B(7, 1)
- d = √[(7-3)² + (1-4)²] = √[16 + 9] = √25 = 5 units
2. Speed, Distance, and Time Relationship
These three quantities form a fundamental relationship in kinematics. The formulas can be rearranged based on what you need to find.
Key Formulas:
- Distance = Speed × Time
- Speed = Distance/Time
- Time = Distance/Speed
Example: A car travels at 60 km/h for 3 hours. Distance = 60 × 3 = 180 km
3. Distance in Uniformly Accelerated Motion
When an object moves with constant acceleration, the distance covered depends on initial velocity, time, and acceleration.
Formula: s = ut + ½at²
Example: A car starting from rest (u = 0) accelerates at 2 m/s² for 5 seconds.
- s = 0(5) + ½(2)(5)² = 0 + ½(2)(25) = 25 meters
4. Euclidean Distance
The generalized distance formula for any number of dimensions. In 2D and 3D, it reduces to the standard distance formulas.
For n dimensions: d = √[(x₁-y₁)² + (x₂-y₂)² + … + (xₙ-yₙ)²]
5. Manhattan Distance
Also called taxicab distance, it measures distance as if you can only travel along grid lines (like city blocks).
Formula: d = |x₁-x₂| + |y₁-y₂|
Example: Distance from (0,0) to (3,4): Manhattan = |3-0| + |4-0| = 7 units (vs. Euclidean = 5 units)
6. Perpendicular Distance from Point to Line
The shortest distance from a point to a line is always perpendicular to that line.
Formula: d = |ax₁ + by₁ + c|/√(a² + b²)
Where the line equation is ax + by + c = 0 and point is (x₁, y₁)
7. Stopping Distance in Vehicle Dynamics
Total stopping distance combines reaction distance (distance traveled during driver’s reaction time) and braking distance (distance traveled while braking).
Total Stopping Distance = Reaction Distance + Braking Distance
- Reaction Distance = Speed × Reaction Time
- Braking Distance = v²/(2μg)
8. Distance of Closest Approach
When two objects are moving, the distance of closest approach is the minimum distance between them at any point in their trajectories.
Approach: Use relative velocity concepts and find when the rate of change of distance becomes zero.
9. Height and Distance Formulas (Trigonometry)
These formulas use trigonometric ratios to solve real-world problems involving heights and distances.
Key Relations:
- Height = Distance × tan(angle of elevation)
- Distance = Height/tan(angle of elevation)
- Distance = Height × cot(angle of elevation)
Conclusion
This comprehensive guide covers all essential distance formulas from basic coordinate geometry to advanced applications in physics and computer science. Students should practice these formulas regularly, understand their derivations, and apply them to various problem types to build strong mathematical foundations.
Frequently Asked Questions (FAQs) about Distance Formula
Q. What is the distance formula and why is it important?
The distance formula is a mathematical equation used to calculate the straight-line distance between two points in a coordinate system. The basic 2D formula is d = √[(x₂ – x₁)² + (y₂ – y₁)²]. It’s important because it forms the foundation for coordinate geometry, is used in navigation systems, computer graphics, physics calculations, and real-world applications like finding the shortest path between two locations. The formula is derived from the Pythagorean theorem and is essential for students from class 10 onwards.
Q. What is the difference between Euclidean distance and Manhattan distance?
Euclidean distance measures the straight-line distance between two points (as the crow flies), calculated using the formula d = √[Σ(xᵢ – yᵢ)²]. Manhattan distance measures the distance as if you can only travel along grid lines (like navigating city blocks), calculated as d = Σ|xᵢ – yᵢ|. For example, between points (0,0) and (3,4), Euclidean distance is 5 units while Manhattan distance is 7 units. Euclidean distance is used in physics and geometry, while Manhattan distance is common in computer science, urban planning, and pathfinding algorithms.
Q. How do you calculate stopping distance for a vehicle?
Stopping distance is the total distance a vehicle travels from the moment a driver perceives a hazard until the vehicle comes to a complete stop. It consists of two components:
- Reaction Distance = Speed × Reaction Time (typically 1.5-2 seconds)
- Braking Distance = v²/(2μg), where v is velocity, μ is the coefficient of friction, and g is gravity
Total Stopping Distance = Reaction Distance + Braking Distance. For example, a car traveling at 20 m/s with a reaction time of 1.5 seconds and friction coefficient of 0.7 would have a stopping distance of approximately 30 + 29 = 59 meters.
Q. What is the formula for distance between two parallel lines?
For two parallel lines in 2D with equations ax + by + c₁ = 0 and ax + by + c₂ = 0, the distance between them is:
d = |c₂ – c₁|/√(a² + b²)
The lines must have the same coefficients for x and y (which makes them parallel). In 3D space, the formula for parallel lines uses vector notation: d = |(a₂ – a₁) · (b₁ × b₂)|/|b₁ × b₂|, where a represents position vectors and b represents direction vectors. This formula is important in class 11-12 for analytical geometry problems.
Q. How are speed, distance, and time related in physics?
Speed, distance, and time form a fundamental relationship in kinematics with three interchangeable formulas:
- Distance = Speed × Time (d = s × t)
- Speed = Distance ÷ Time (s = d/t)
- Time = Distance ÷ Speed (t = d/s)
For uniform motion, distance traveled is directly proportional to both speed and time. For non-uniform motion with acceleration, use s = ut + ½at² or v² = u² + 2as. These formulas are essential for solving problems in classes 9-12 involving motion, transportation, and everyday scenarios like calculating travel time or fuel consumption.
Q. What is the distance of closest approach and how is it calculated?
The distance of closest approach is the minimum distance between two moving objects during their motion. It’s particularly important in physics for analyzing collisions, planetary motion, and particle interactions. To calculate it:
- Express the positions of both objects as functions of time
- Find the distance between them as a function of time
- Minimize this distance by taking its derivative and setting it to zero
- Solve for the time at which closest approach occurs
- Substitute this time back to find the minimum distance
The formula often involves relative velocity: d_min = |d₀ × v_rel|/|v_rel|, where d₀ is initial separation and v_rel is relative velocity vector.
Q. How do you solve height and distance problems using trigonometry?
Height and distance problems use trigonometric ratios (sine, cosine, tangent) along with angles of elevation and depression. Key steps:
- Draw a diagram showing the situation
- Identify the angle of elevation (looking up) or depression (looking down)
- Apply appropriate formula:
- Height = Distance × tan(angle) when you know horizontal distance
- Distance = Height/tan(angle) when you know height
- Use sine/cosine for problems involving slant distance
Example: A tower’s angle of elevation from a point 50m away is 30°. Height = 50 × tan(30°) = 50 × (1/√3) ≈ 28.87 meters. These problems are common in class 10 and have practical applications in surveying, navigation, and architecture.