Conversion Factors
A conversion factor is a ratio equal to 1, such as 12 inches per 1 foot.
Free Interactive Math & Science Tool
Unit Conversions & Dimensional Analysis
Learn to convert units by building conversion factors, canceling units, and following the units all the way to the final answer. Practice simple conversions, multi-step dimensional analysis, rates, metric units, area, volume, and science applications.
Start practicingNeed Help With Dimensional Analysis?
Learn how conversion factors work, why units cancel, how to build multi-step conversion chains, and how to handle rates, squared units, and cubic units.
Read the Dimensional Analysis Guide
Practice Mode
Practice familiar one-step unit conversions.
Problem 1 of 10
10%
Current Problem
Identify the starting and target units.
One-Step
Beginner
Convert
3 ft → ? in
Convert 3 feet to inches using dimensional analysis.
Start
3 ft
Target Unit
in
Current Units
ft
1
Build Chain
2
Calculate
Step 1
Identify the units
Determine which unit needs to cancel and which unit must remain in the final answer.
Current Problem
3 ft → ? in
Dimensional Analysis Workbench
Build a conversion chain that cancels feet.
Build your chain
Unit Check
ft → ?
Choose a conversion factor that places ft opposite the existing ft so those units cancel.
Conversion Factors
Choose the factor in the correct orientation.
Area & Volume Rule
The conversion factor must match the power of the unit.
1 ft² =
(12 in)² =
144 in²
Because the unit is squared, both the numerical conversion and the unit are squared.
Compound Unit
Numerator and denominator units must both be handled.
mi/hr → ft/s
Convert miles to feet and hours to seconds so the final units become feet per second.
Supplied Relationship
Use the relationship as a conversion factor.
1 g = 1000 mg
Any equality can generate two reciprocal conversion factors.
Your Work
Which unit should cancel first?
Completed Conversion
3 ft × 12 in / 1 ft = 36 in
Start by identifying which unit must cancel.
Hint
Put the unwanted unit on the opposite side of the fraction so it cancels.
Quick Reference
Common Unit Conversions
Use these equalities to build conversion factors. Each equality can be written in either orientation.
Length
| Equality | Conversion Factor 1 | Conversion Factor 2 |
|---|---|---|
| 1 ft = 12 in | 12 in / 1 ft | 1 ft / 12 in |
| 1 yd = 3 ft | 3 ft / 1 yd | 1 yd / 3 ft |
| 1 mi = 5280 ft | 5280 ft / 1 mi | 1 mi / 5280 ft |
| 1 in = 2.54 cm | 2.54 cm / 1 in | 1 in / 2.54 cm |
| 1 m = 100 cm | 100 cm / 1 m | 1 m / 100 cm |
| 1 km = 1000 m | 1000 m / 1 km | 1 km / 1000 m |
Mass & Weight
| Equality | Conversion Factor 1 | Conversion Factor 2 |
|---|---|---|
| 1 lb = 16 oz | 16 oz / 1 lb | 1 lb / 16 oz |
| 1 ton = 2000 lb | 2000 lb / 1 ton | 1 ton / 2000 lb |
| 1 kg = 1000 g | 1000 g / 1 kg | 1 kg / 1000 g |
| 1 g = 1000 mg | 1000 mg / 1 g | 1 g / 1000 mg |
Volume & Capacity
| Equality | Conversion Factor 1 | Conversion Factor 2 |
|---|---|---|
| 3 tsp = 1 tbsp | 1 tbsp / 3 tsp | 3 tsp / 1 tbsp |
| 2 tbsp = 1 fl oz | 1 fl oz / 2 tbsp | 2 tbsp / 1 fl oz |
| 8 fl oz = 1 cup | 1 cup / 8 fl oz | 8 fl oz / 1 cup |
| 2 cups = 1 pint | 1 pint / 2 cups | 2 cups / 1 pint |
| 2 pints = 1 quart | 1 quart / 2 pints | 2 pints / 1 quart |
| 4 quarts = 1 gallon | 1 gallon / 4 quarts | 4 quarts / 1 gallon |
| 1 L = 1000 mL | 1000 mL / 1 L | 1 L / 1000 mL |
Time
| Equality | Conversion Factor 1 | Conversion Factor 2 |
|---|---|---|
| 60 sec = 1 min | 1 min / 60 sec | 60 sec / 1 min |
| 60 min = 1 hr | 1 hr / 60 min | 60 min / 1 hr |
| 24 hr = 1 day | 1 day / 24 hr | 24 hr / 1 day |
| 7 days = 1 week | 1 week / 7 days | 7 days / 1 week |
Metric Prefixes
| Prefix | Symbol | Factor |
|---|---|---|
| Giga | G | 10⁹ |
| Mega | M | 10⁶ |
| Kilo | k | 10³ |
| Hecto | h | 10² |
| Deka | da | 10¹ |
| Base unit | — | 10⁰ |
| Deci | d | 10⁻¹ |
| Centi | c | 10⁻² |
| Milli | m | 10⁻³ |
| Micro | µ | 10⁻⁶ |
| Nano | n | 10⁻⁹ |
Area Conversions
Square the entire linear conversion factor.
Since 1 ft = 12 in: 1 ft² = 12² in² = 144 in².
| Equality | Equivalent Area |
|---|---|
| 1 ft² | 144 in² |
| 1 yd² | 9 ft² |
| 1 m² | 10,000 cm² |
| 1 km² | 1,000,000 m² |
Cubic Unit Conversions
Cube the entire linear conversion factor.
Since 1 ft = 12 in: 1 ft³ = 12³ in³ = 1728 in³.
| Equality | Equivalent Volume |
|---|---|
| 1 ft³ | 1728 in³ |
| 1 yd³ | 27 ft³ |
| 1 m³ | 1,000,000 cm³ |
| 1 cm³ | 1 mL |
Common Rate Relationships
| Relationship | Useful For |
|---|---|
| 1 mi = 5280 ft | mi/hr ↔ ft/hr |
| 1 hr = 3600 sec | per hour ↔ per second |
| 1 km = 1000 m | km/hr ↔ m/hr |
| 1 hr = 60 min | per hour ↔ per minute |
Problem Complete
3 ft = 36 in
Feet canceled, inches remained, and the numerical conversion gave 36 inches.
Problems Solved
0
Correct Steps
0
Incorrect Attempts
0
Current Streak
0
Practice Round Complete
Great work!
You completed all 10 dimensional analysis problems.
Problems Solved
0
Correct Steps
0
Incorrect Attempts
0
Step Accuracy
0%
Cancel Units
Place matching units in opposite positions so they cancel just like numerical factors.
Build Chains
Multi-step conversions are solved by connecting several correct conversion factors in sequence.
Follow the Target Unit
Your setup is structurally correct when every unwanted unit cancels and only the desired unit remains.
Complete Example
Convert 60 mi/hr to ft/s
Both parts of the rate must be converted: miles to feet and hours to seconds.
60 mi
1 hr
×
1 hr
5280 ft
1 mi
×
1 mi
1 hr
3600 s
3600 s
mi
cancels
hr
cancels
60 × 5280 ÷ 3600 = 88 ft/s
The numerical answer is 88, but the unit analysis is what confirms that the setup produces feet per second.
Practice Unit Conversions With Dimensional Analysis
Dimensional analysis is a systematic method for converting between units. Instead of memorizing separate procedures for every conversion, students use ratios called conversion factors and arrange them so unwanted units cancel.
This interactive dimensional analysis practice tool focuses on the setup itself. Students identify target units, choose conversion factors, orient each factor correctly, watch units cancel, and then calculate the final numerical answer.
One-Step Unit Conversions
One-step conversions use a single relationship such as 1 foot = 12 inches. The conversion factor must be oriented so the starting unit cancels and the desired unit remains.
Multi-Step Dimensional Analysis
More advanced conversions may require several factors. For example, converting miles per hour to feet per second requires both a distance conversion and a time conversion.
Metric Unit Conversions
Metric conversions use powers of ten and prefixes such as kilo-, centi-, milli-, micro-, and nano-. Dimensional analysis provides a consistent method even when several prefixes are involved.
Rates and Compound Units
Units such as miles per hour, dollars per pound, meters per second, or milligrams per milliliter contain both a numerator and denominator. Each part of the compound unit may need its own conversion.
Area and Volume Conversions
Squared and cubed units require special attention. If 1 foot = 12 inches, then 1 square foot equals 144 square inches and 1 cubic foot equals 1728 cubic inches because the conversion factor must also be squared or cubed.
Science Applications
Dimensional analysis is widely used in chemistry, physics, engineering, health sciences, and other fields because it allows unfamiliar quantities to be converted using supplied relationships while preserving unit consistency.
Built-In Conversion Reference
The tool includes a searchable reference chart with common length, mass, volume, time, metric, area, cubic, and rate conversions. Each equality can be viewed as two reciprocal conversion factors for use in dimensional analysis.
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