Cinema Calculations Handbook

Introduction to Cinema Mathematics

Professional cinematography relies on precise mathematical calculations that separate amateur work from professional results. Understanding these formulas allows cinematographers to make informed creative decisions while maintaining technical excellence.

Why Cinema Math Matters

Every great cinematographer understands the mathematical relationships between aperture, focal length, distance, and sensor size. These calculations determine depth of field, field of view, exposure values, and ultimately the visual language of your film. Mastering these formulas gives you creative control over every aspect of the image.

This comprehensive handbook covers the essential mathematical concepts that form the foundation of professional cinematography, from basic optical calculations to advanced exposure mathematics.

1. Depth of Field Calculations

Depth of field is one of the most powerful creative tools in cinematography. Understanding the mathematical relationships allows precise control over what's in focus and what's not.

Circle of Confusion

Formula

CoC = Sensor Width ÷ (Crop Factor × 1500)

Calculate the circle of confusion for your specific camera sensor.

Where:

Sensor Width= Physical width of the sensor(mm)
Crop Factor= Sensor crop factor relative to full frame(multiplier)

Example: Full Frame Camera (36mm sensor)

Calculation: 36 ÷ (1.0 × 1500) = 0.024mm

Result: 0.024mm circle of confusion

Depth of Field Formula

Formula

DoF = (2 × N × CoC × s²) ÷ (f² - N × CoC × s)

Calculate the total depth of field for any lens and camera combination.

Where:

N= F-stop (aperture)(f-number)
CoC= Circle of confusion(mm)
s= Subject distance(mm)
f= Focal length(mm)

Example: 85mm lens, f/2.8, 3m subject distance

Calculation: (2 × 2.8 × 0.024 × 3000²) ÷ (85² - 2.8 × 0.024 × 3000)

Result: ≈ 180mm total DoF

Common Circle of Confusion Values

Full Frame (35mm): 0.024mm
APS-C Canon: 0.015mm
APS-C Sony/Nikon: 0.016mm
Micro Four Thirds: 0.012mm
Super 35mm Film: 0.025mm

Near and Far Focus Points

Formula

Near = s × f² ÷ (f² + N × CoC × s) | Far = s × f² ÷ (f² - N × CoC × s)

Calculate the exact near and far focus limits.

Where:

s= Focus distance(mm)
f= Focal length(mm)
N= F-stop(f-number)
CoC= Circle of confusion(mm)

Example: 50mm lens, f/4, focused at 2m

Calculation: Near: 1.85m | Far: 2.17m

Result: 320mm depth of field

Depth of Field Reference Table

Focal Length	f/1.4	f/2.8	f/5.6
35mm @ 3m	0.8m	1.6m	3.2m
85mm @ 3m	0.08m	0.16m	0.32m
135mm @ 3m	0.03m	0.06m	0.12m

2. Hyperfocal Distance

Hyperfocal distance is the closest focusing distance where objects at infinity appear acceptably sharp. It's crucial for landscape cinematography and maximizing depth of field.

Hyperfocal Distance Formula

Formula

H = f² ÷ (N × CoC)

Calculate the hyperfocal distance for maximum depth of field.

Where:

f= Focal length(mm)
N= F-stop (aperture)(f-number)
CoC= Circle of confusion(mm)

Example: 24mm lens at f/8 on full frame

Calculation: 24² ÷ (8 × 0.024) = 576 ÷ 0.192

Result: 3m hyperfocal distance

Near Limit When Focused at Hyperfocal

Formula

Near Limit = H ÷ 2

When focused at hyperfocal distance, everything from half that distance to infinity is sharp.

Where:

H= Hyperfocal distance(mm)

Example: Hyperfocal distance of 3m

Calculation: 3000 ÷ 2 = 1500mm

Result: Sharp from 1.5m to infinity

Hyperfocal Distance Applications

Landscape shots: Maximum sharpness from foreground to background
Wide establishing shots: Ensure entire scene is in focus
Documentary work: Maintain focus without constant adjustment
Action sequences: Keep moving subjects in acceptable focus

Hyperfocal Distance Reference (Full Frame)

Focal Length	f/2.8	f/5.6	f/11
14mm	2.9m	1.4m	0.7m
24mm	8.6m	4.3m	2.2m
35mm	18.2m	9.1m	4.6m

3. Field of View & Angle Calculations

Field of view calculations help determine lens choice, camera placement, and framing decisions. Understanding these relationships is essential for shot planning and lens selection.

Horizontal Field of View

Formula

HFoV = 2 × arctan(Sensor Width ÷ (2 × Focal Length))

Calculate the horizontal field of view angle for any lens and sensor combination.

Where:

Sensor Width= Physical width of sensor(mm)
Focal Length= Lens focal length(mm)

Example: 50mm lens on full frame (36mm sensor)

Calculation: 2 × arctan(36 ÷ (2 × 50)) = 2 × arctan(0.36)

Result: 39.6° horizontal field of view

Subject Width in Frame

Formula

Subject Width = 2 × Distance × tan(HFoV ÷ 2)

Calculate how much of the scene will fit in frame at a given distance.

Where:

Distance= Distance to subject(mm)
HFoV= Horizontal field of view(degrees)

Example: 50mm lens, 3m from subject

Calculation: 2 × 3000 × tan(39.6° ÷ 2) = 2 × 3000 × tan(19.8°)

Result: 2.16m subject width in frame

Distance for Desired Framing

Formula

Distance = (Subject Width ÷ 2) ÷ tan(HFoV ÷ 2)

Calculate camera distance needed to frame a subject of known width.

Where:

Subject Width= Width of subject to frame(mm)
HFoV= Horizontal field of view(degrees)

Example: Frame 2m wide subject with 50mm lens

Calculation: (2000 ÷ 2) ÷ tan(39.6° ÷ 2) = 1000 ÷ tan(19.8°)

Result: 2.78m camera distance needed

Common Focal Length Classifications

Ultra Wide (14-24mm): 114°-84° field of view
Wide Angle (24-35mm): 84°-63° field of view
Normal (35-85mm): 63°-28° field of view
Telephoto (85-200mm): 28°-12° field of view
Super Telephoto (200mm+): 12°+ field of view

Field of View Reference (Full Frame)

Focal Length	Horizontal FoV	Subject Width @ 3m
24mm	84.1°	4.5m
50mm	39.6°	2.16m
85mm	23.9°	1.27m

4. Lens Mathematics

Understanding lens mathematics helps with focal length conversions, crop factor calculations, and equivalent lens comparisons across different sensor formats.

Crop Factor Calculation

Formula

Crop Factor = Full Frame Diagonal ÷ Sensor Diagonal

Calculate the crop factor for any sensor size relative to full frame.

Where:

Full Frame Diagonal= Always 43.27mm(mm)
Sensor Diagonal= Diagonal of your sensor(mm)

Example: APS-C sensor (23.6 × 15.6mm)

Calculation: 43.27 ÷ √(23.6² + 15.6²) = 43.27 ÷ 28.4

Result: 1.52x crop factor

Equivalent Focal Length

Formula

Equivalent FL = Actual FL × Crop Factor

Calculate the full-frame equivalent focal length for cropped sensors.

Where:

Actual FL= Physical focal length of lens(mm)
Crop Factor= Sensor crop factor(multiplier)

Example: 50mm lens on APS-C (1.5x crop)

Calculation: 50 × 1.5 = 75mm

Result: 75mm full-frame equivalent

Magnification Factor

Formula

Magnification = Image Distance ÷ Object Distance

Calculate the magnification ratio for macro and close-up work.

Where:

Image Distance= Distance from lens to sensor(mm)
Object Distance= Distance from lens to subject(mm)

Example: Macro lens: 100mm from sensor, 200mm to subject

Calculation: 100 ÷ 200 = 0.5

Result: 0.5x magnification (1:2 ratio)

Crop Factor Applications

Lens selection: Choose lenses based on equivalent field of view
Depth of field: Smaller sensors have greater apparent DoF
Low light: Larger sensors generally perform better
Telephoto reach: Crop sensors extend telephoto range

Common Sensor Formats

Format	Dimensions	Crop Factor	50mm Equivalent
Full Frame	36 × 24mm	1.0x	50mm
APS-C Canon	22.3 × 14.9mm	1.6x	80mm
Micro 4/3	17.3 × 13mm	2.0x	100mm

5. Exposure Mathematics

Exposure calculations form the foundation of cinematography. Understanding the mathematical relationships between aperture, shutter speed, and ISO allows precise control over image exposure.

Exposure Value (EV)

Formula

EV = log₂(N² ÷ t)

Calculate the exposure value for any aperture and shutter speed combination.

Where:

N= F-stop (aperture)(f-number)
t= Shutter speed(seconds)

Example: f/4 at 1/60 second

Calculation: log₂(4² ÷ (1/60)) = log₂(16 × 60) = log₂(960)

Result: EV ≈ 9.9

Stop Difference Calculation

Formula

Stop Difference = log₂(Light₁ ÷ Light₂)

Calculate the difference in stops between two light levels.

Where:

Light₁= First light measurement(lux or fc)
Light₂= Second light measurement(lux or fc)

Example: Comparing 1000 lux to 250 lux

Calculation: log₂(1000 ÷ 250) = log₂(4)

Result: 2 stops difference

T-Stop to F-Stop Conversion

Formula

T-Stop = F-Stop ÷ √(Light Transmission)

Convert between f-stops and t-stops accounting for lens light transmission.

Where:

F-Stop= Geometric aperture(f-number)
Light Transmission= Lens transmission efficiency (0.7-0.95)(percentage)

Example: f/2.8 lens with 85% transmission

Calculation: 2.8 ÷ √(0.85) = 2.8 ÷ 0.922

Result: T3.0 actual light transmission

Cinema vs Photography Exposure

Cinema exposure differs from photography in several key ways:

Shutter angle: 180° = 1/(2×frame rate) shutter speed
T-stops: More accurate than f-stops for consistent exposure
False color: Use waveforms and false color for precise exposure
Log recording: Requires different exposure strategies

Shutter Angle to Shutter Speed Conversion

Frame Rate	180° Shutter	90° Shutter	360° Shutter
24fps	1/48s	1/96s	1/24s
30fps	1/60s	1/120s	1/30s
60fps	1/120s	1/240s	1/60s

6. Color Temperature & Light

Color temperature calculations help match different light sources and achieve consistent color reproduction across various lighting conditions.

Mired Calculation

Formula

Mired = 1,000,000 ÷ Color Temperature (K)

Convert color temperature to mireds for filter calculations.

Where:

Color Temperature= Light source temperature(Kelvin)

Example: Daylight at 5600K

Calculation: 1,000,000 ÷ 5600 = 178.6

Result: 179 mireds

Filter Strength Calculation

Formula

Filter Mireds = Source Mireds - Target Mireds

Calculate the required filter strength to match color temperatures.

Where:

Source Mireds= Current light source in mireds(mireds)
Target Mireds= Desired color temperature in mireds(mireds)

Example: Tungsten (3200K) to Daylight (5600K)

Calculation: 312 - 179 = +133 mireds (Full CTB)

Result: +133 mired filter (CTB)

Light Falloff (Inverse Square Law)

Formula

Light Intensity = Source Intensity ÷ Distance²

Calculate how light intensity decreases with distance.

Where:

Source Intensity= Light output at source(lux or fc)
Distance= Distance from light source(meters)

Example: 1000W light at 2m vs 4m

Calculation: At 2m: 100% | At 4m: 100% ÷ (4²÷2²) = 25%

Result: 4x less light at double distance

Common Color Temperatures

Candlelight: 1900K (526 mireds)
Tungsten: 3200K (312 mireds)
Fluorescent: 4100K (244 mireds)
Daylight: 5600K (179 mireds)
Overcast Sky: 6500K (154 mireds)
Blue Sky: 10000K (100 mireds)

Case Study: Mixed Lighting Scenario

CommercialAdvanced

4 hours setup

Scenario

Interior scene with tungsten practicals and daylight from windows, requiring color consistency for a commercial shoot.

Challenge

Balance tungsten (3200K) and daylight (5600K) sources while maintaining natural-looking lighting and proper exposure levels.

Solution

1Measure existing light levels and color temperatures
2Calculate filter requirements for color matching
3Determine additional lighting needs using inverse square law
4Plan gel and diffusion strategy
5Test exposure and color balance

Calculations:

Window light: 5600K (179 mireds)Tungsten practicals: 3200K (312 mireds)Required correction: 312 - 179 = +133 mireds (Full CTB)Light falloff: Window at 3m = 100%, Practical at 1m = 900% relativeND required: log₂(900/100) = 3.17 stops ND on practicals

Result

Achieved consistent 5600K color temperature throughout scene with natural-looking lighting balance and proper exposure ratios.

Key Takeaways

Mired calculations simplify color temperature matching
Inverse square law is crucial for balancing multiple sources
Always measure and calculate rather than guess
Test shots save time and ensure consistency

Mastering Cinema Mathematics

These mathematical foundations give you precise control over every aspect of the cinematic image. From depth of field to color temperature, understanding these relationships allows you to make informed creative decisions while maintaining technical excellence.

Practice these calculations until they become second nature, and consider using professional tools like CineMath to handle complex calculations quickly and accurately on set.

Master Cinema Mathematics with CineMath

Get instant cinema calculations with professional accuracy. Focus on your craft while CineMath handles the math.

Download CineMath

Table of Contents

Introduction to Cinema Mathematics

Why Cinema Math Matters

1. Depth of Field Calculations

Circle of Confusion

Where:

Example: Full Frame Camera (36mm sensor)

Depth of Field Formula

Where:

Example: 85mm lens, f/2.8, 3m subject distance

Common Circle of Confusion Values

Near and Far Focus Points

Where:

Example: 50mm lens, f/4, focused at 2m

Depth of Field Reference Table

2. Hyperfocal Distance

Hyperfocal Distance Formula

Where:

Example: 24mm lens at f/8 on full frame

Near Limit When Focused at Hyperfocal

Where:

Example: Hyperfocal distance of 3m

Hyperfocal Distance Applications

Hyperfocal Distance Reference (Full Frame)

3. Field of View & Angle Calculations

Horizontal Field of View

Where:

Example: 50mm lens on full frame (36mm sensor)

Subject Width in Frame

Where:

Example: 50mm lens, 3m from subject

Distance for Desired Framing

Where:

Example: Frame 2m wide subject with 50mm lens

Common Focal Length Classifications

Field of View Reference (Full Frame)

4. Lens Mathematics

Crop Factor Calculation

Where:

Example: APS-C sensor (23.6 × 15.6mm)

Equivalent Focal Length

Where:

Example: 50mm lens on APS-C (1.5x crop)

Magnification Factor

Where:

Example: Macro lens: 100mm from sensor, 200mm to subject

Crop Factor Applications

Common Sensor Formats

5. Exposure Mathematics

Exposure Value (EV)

Where:

Example: f/4 at 1/60 second

Stop Difference Calculation

Where:

Example: Comparing 1000 lux to 250 lux

T-Stop to F-Stop Conversion

Where:

Example: f/2.8 lens with 85% transmission

Cinema vs Photography Exposure

Shutter Angle to Shutter Speed Conversion

6. Color Temperature & Light

Mired Calculation

Where:

Example: Daylight at 5600K

Filter Strength Calculation

Where:

Example: Tungsten (3200K) to Daylight (5600K)

Light Falloff (Inverse Square Law)

Where:

Example: 1000W light at 2m vs 4m

Common Color Temperatures

Case Study: Mixed Lighting Scenario

Scenario

Challenge

Solution

Calculations:

Key Takeaways

Mastering Cinema Mathematics

Master Cinema Mathematics with CineMath