Cinema Calculations Handbook

Updated

Depth of field, hyperfocal distance, angle of view, crop factors, exposure, ND and mireds: the formulas camera and lighting departments use, with worked examples and reference tables.

35 min readAdvanced
  • Cinematography
  • Depth of Field
  • Exposure
  • Color Temperature
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Introduction to Cinema Mathematics

Focus, framing, exposure and color all come down to a handful of formulas. This handbook sets out the ones camera and lighting departments use on set, with each worked example and table computed from the formula shown beside it.

1. Depth of Field Calculations

Depth of field is the zone in front of and behind the focus point that still looks sharp. It depends on four things: focal length, stop, focus distance and the circle of confusion (CoC), the largest blur spot that still reads as a point at the intended viewing size. The depth of field guide adds Super 35 charts in feet and meters.

Circle of Confusion
CoC = Sensor Diagonal ÷ 1500

The usual starting point for the circle of confusion. It assumes the image is enlarged to about 30cm across and viewed from reading distance; a big theatrical screen or a tight crop in post calls for a smaller value.

Where

Sensor Diagonal
√(width² + height²) of the area you record(mm)

Example: Full frame (36 × 24mm, 43.3mm diagonal)

Calculation:
43.27 ÷ 1500 = 0.029mm
Result:
0.029mm, usually rounded to 0.030mm
Near and Far Focus Limits
Near = s × f² ÷ (f² + N × CoC × (s − f)) | Far = s × f² ÷ (f² − N × CoC × (s − f))

The exact thin-lens limits, as CineMath computes them. When the far denominator reaches zero or below, the far limit is infinity: you are focused at or past the hyperfocal distance.

Where

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

Example: 50mm at f/4, focused at 2m, full frame (CoC 0.030mm)

Calculation:
Near = 2000 × 2500 ÷ (2500 + 234) = 1.83m | Far = 2000 × 2500 ÷ (2500 − 234) = 2.21m
Result:
Sharp from 1.83m to 2.21m: about 0.38m of depth of field
Total Depth of Field
DoF = Far − Near ≈ 2 × N × CoC × s² ÷ f²

Subtract the limits for the exact figure. The shortcut on the right is close as long as the focus distance is well inside the hyperfocal distance, and it shows the relationships at a glance: depth of field grows with the stop and the square of the distance, and shrinks with the square of the focal length.

Where

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

Example: 85mm at f/2.8, focused at 3m, full frame (CoC 0.030mm)

Calculation:
Exact: 3.105m − 2.902m = 0.204m | Shortcut: 2 × 2.8 × 0.030 × 3000² ÷ 85² = 209mm
Result:
About 20cm of depth of field

Depth of Field Reference Table

Total depth of field at 3m on full frame, CoC 0.030mm, from the exact near and far limits above.

Lens and focusf/1.4f/2.8f/5.6
35mm @ 3m0.62m1.27m2.92m
85mm @ 3m0.10m0.20m0.41m
135mm @ 3m0.04m0.08m0.16m

2. Hyperfocal Distance

The hyperfocal distance is the closest focus distance at which the far limit reaches infinity. Focus there and everything from half that distance to the horizon is acceptably sharp, which is why it is the standard setting for wide, deep shots and for run-and-gun work where nobody is pulling focus. The hyperfocal distance guide has charts for five formats, and the free hyperfocal distance chart is a printable PDF for Super 35, full frame and large format.

Hyperfocal Distance Formula
H = f² ÷ (N × CoC) + f

The + f term matters only with short focal lengths at close distances; many tables drop it. CineMath includes it.

Where

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

Example: 24mm at f/8 on full frame (CoC 0.030mm)

Calculation:
24² ÷ (8 × 0.030) + 24 = 576 ÷ 0.24 + 24 = 2,424mm
Result:
Hyperfocal distance about 2.4m
Near Limit When Focused at Hyperfocal
Near Limit = H ÷ 2

Focused at the hyperfocal distance, sharpness runs from half that distance to infinity.

Where

H
Hyperfocal distance(mm)

Example: Hyperfocal distance of 2.42m

Calculation:
2,424 ÷ 2 = 1,212mm
Result:
Sharp from about 1.2m to infinity

Hyperfocal Distance Reference (Full Frame, CoC 0.030mm)

Focal Lengthf/2.8f/5.6f/11
14mm2.3m1.2m0.6m
24mm6.9m3.5m1.8m
35mm14.6m7.3m3.7m

3. Field of View & Angle Calculations

Angle of view depends only on focal length and the size of the recorded image. The same formula gives the horizontal, vertical or diagonal angle: use the sensor's width, height or diagonal. The field of view guide covers angle and frame size in more depth, and the free field of view calculator works them out in your browser.

Angle of View
AoV = 2 × arctan(Sensor Dimension ÷ (2 × Focal Length))

Use the width for the horizontal angle, the height for the vertical angle and the diagonal for the diagonal angle. Lens makers usually quote the diagonal.

Where

Sensor Dimension
Width, height or diagonal of the recorded area(mm)
Focal Length
Lens focal length(mm)

Example: 50mm lens on full frame (36 × 24mm)

Calculation:
Horizontal 2 × arctan(36 ÷ 100) | Vertical 2 × arctan(24 ÷ 100) | Diagonal 2 × arctan(43.27 ÷ 100)
Result:
39.6° horizontal, 27.0° vertical, 46.8° diagonal
Frame Width at a Distance
Frame Width = Distance × Sensor Width ÷ Focal Length

How much of the scene fits across the frame. This is the same as 2 × Distance × tan(HFoV ÷ 2); swap in the sensor height for frame height.

Where

Distance
Distance from the camera to the subject plane(mm)
Sensor Width
Width of the recorded area(mm)
Focal Length
Lens focal length(mm)

Example: 50mm lens on full frame, 3m from the subject

Calculation:
3000 × 36 ÷ 50 = 2,160mm
Result:
The frame is 2.16m wide (and 1.44m tall)
Distance for Desired Framing
Distance = Subject Width × Focal Length ÷ Sensor Width

The camera distance needed to fit a subject of known width across the frame.

Where

Subject Width
Width to fit in frame(mm)
Focal Length
Lens focal length(mm)
Sensor Width
Width of the recorded area(mm)

Example: Fit a 2m-wide subject with a 50mm lens on full frame

Calculation:
2000 × 50 ÷ 36 = 2,778mm
Result:
Place the camera about 2.78m away

Field of View Reference (Full Frame, 36 × 24mm)

Focal LengthHorizontalVerticalDiagonalFrame @ 3m (W × H)
24mm73.7°53.1°84.1°4.50 × 3.00m
35mm54.4°37.8°63.4°3.09 × 2.06m
50mm39.6°27.0°46.8°2.16 × 1.44m
85mm23.9°16.1°28.6°1.27 × 0.85m
135mm15.2°10.2°18.2°0.80 × 0.53m

4. Lens Mathematics

Crop factors let you compare lenses across sensor sizes. There are two conventions, and they disagree whenever the two formats have different shapes. The free focal length match calculator matches lenses by width.

Crop Factor (Diagonal)
Crop Factor = 43.27 ÷ Sensor Diagonal

The stills convention: 43.27mm is the diagonal of the 36 × 24mm full-frame format.

Where

43.27
Full-frame diagonal, √(36² + 24²)(mm)
Sensor Diagonal
Diagonal of your sensor(mm)

Example: APS-C sensor (23.5 × 15.6mm)

Calculation:
43.27 ÷ √(23.5² + 15.6²) = 43.27 ÷ 28.21
Result:
1.53x crop factor
Crop Factor (Width) and Focal Length Match
Matching Focal Length = Focal Length × (Target Width ÷ Original Width)

The usual convention on a film set, because shots are framed and matched on the horizontal angle. CineMath's Focal Length Match calculator works this way.

Where

Focal Length
The lens on the original camera(mm)
Original Width
Sensor width of the original camera(mm)
Target Width
Sensor width of the other camera(mm)

Example: 50mm on full frame, matched on Super 35 (24.89mm wide)

Calculation:
50 × 24.89 ÷ 36 = 34.6mm (width crop factor 36 ÷ 24.89 = 1.45)
Result:
A 35mm lens on Super 35 gives almost the same horizontal view as a 50mm on full frame
Magnification
Magnification = Image Size ÷ Subject Size

How large the subject is on the sensor compared with life. 1:1 (1x) is true macro.

Where

Image Size
Size of the subject on the sensor(mm)
Subject Size
Real size of the subject(mm)

Example: A 72mm-wide watch face filling a 36mm-wide sensor

Calculation:
36 ÷ 72 = 0.5
Result:
0.5x magnification (1:2)

Common Sensor Formats

FormatDimensionsCrop (diagonal)Crop (width)Matches a 50mm on full frame (horizontal)
Full frame36 × 24mm1.0x1.0x50mm
Super 35 (4-perf film gate)24.89 × 18.66mm1.39x1.45x34.6mm
APS-C, Sony and Nikon23.5 × 15.6mm1.53x1.53x32.6mm
APS-C, Canon22.3 × 14.9mm1.61x1.61x31.0mm
Micro Four Thirds17.3 × 13mm2.0x2.08x24.0mm

Digital "Super 35" sensors vary from about 23 to 28mm wide, so check the recorded width of your camera and mode. The sensor size chart lists them for current cinema cameras.

5. Exposure Mathematics

Exposure is counted in stops, and every stop is a factor of two in light. Aperture, shutter, ISO and ND each move exposure by a known number of stops, so they can be traded against one another. The exposure triangle guide covers those trades on a cine camera, and the exposure value chart turns a meter EV reading into a T-stop.

Exposure Value (EV)
EV = log₂(N² ÷ t)

One number for every aperture and shutter pair that gives the same exposure. Each step of 1 EV is one stop.

Where

N
F-number (aperture)(f-number)
t
Exposure time(seconds)

Example: f/4 at 1/60 second

Calculation:
log₂(4² ÷ (1/60)) = log₂(16 × 60) = log₂(960)
Result:
EV ≈ 9.9
Stops Between Two Readings
Stops = log₂(Light₁ ÷ Light₂) = 2 × log₂(N₁ ÷ N₂)

Use the first form for illuminance readings and the second for two stops read off a meter.

Where

Light₁, Light₂
Two illuminance readings(lux or fc)
N₁, N₂
Two f- or T-numbers(f-number)

Example: 1000 lux vs 250 lux, and f/8 vs T2.8

Calculation:
log₂(1000 ÷ 250) = 2 | 2 × log₂(8 ÷ 2.8) = 3.0
Result:
2 stops, and 3 stops
F-Stop to T-Stop
T-Stop = F-Stop ÷ √(Transmittance)

A T-stop is the f-number corrected for the light the lens absorbs and scatters. Cine lenses are calibrated in T-stops so that every lens in a set gives the same exposure at the same marking.

Where

F-Stop
Geometric aperture(f-number)
Transmittance
Fraction of light the lens passes, as a decimal(0 to 1)

Example: f/2.8 lens that transmits 85% of the light

Calculation:
2.8 ÷ √0.85 = 2.8 ÷ 0.922 = 3.04
Result:
About T3, a quarter of a stop darker than f/2.8
Shutter Angle to Exposure Time
Exposure Time = Shutter Angle ÷ (360 × Frame Rate)

A 180° shutter exposes each frame for half the frame interval, which gives the motion blur audiences read as normal.

Where

Shutter Angle
Open portion of the rotating shutter(degrees)
Frame Rate
Capture frame rate(fps)

Example: 180° at 24fps

Calculation:
180 ÷ (360 × 24) = 1/48
Result:
1/48 second

Shutter Angle to Shutter Speed Conversion

Frame Rate360° Shutter180° Shutter90° Shutter
24fps1/24s1/48s1/96s
25fps1/25s1/50s1/100s
30fps1/30s1/60s1/120s
60fps1/60s1/120s1/240s
ND Filter Strength
Stops = Optical Density ÷ 0.3

Each 0.3 of density halves the light (the exact figure is log₁₀2 = 0.301). Light passed = 10^−density.

Where

Optical Density
The number on the filter, such as ND 0.9(density)

Example: The meter reads f/8 and you want to shoot at T2.8

Calculation:
2 × log₂(8 ÷ 2.8) = 3 stops; 3 × 0.3 = 0.9
Result:
Add ND 0.9
ND DensityStopsLight Passed
ND 0.3150%
ND 0.6225%
ND 0.9312.6%
ND 1.246.3%
ND 1.553.2%
ND 1.861.6%
ND 2.170.8%
ND 2.480.4%

6. Color Temperature & Light

Correction gels are rated in mireds rather than kelvin, because a gel shifts every source by the same number of mireds but by a different number of kelvin. The same mired math is used in color grading.

Mired Calculation
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 Mired Shift
Mired Shift = Target Mireds − Source Mireds

Blue gels raise the color temperature and have negative shifts; orange gels lower it and have positive shifts. Shifts add, so two gels together shift by the sum of their values.

Where

Source Mireds
The light as it is(mireds)
Target Mireds
The color temperature you want(mireds)

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

Calculation:
178.6 − 312.5 = −133.9 mireds
Result:
About −134 mireds: a full CTB (Rosco 3202 is −131, LEE 201 is −137)

Published Correction Gel Values

From the Rosco Cinegel Filter Facts and LEE Filters product pages. Correction gels cost light as well as shifting color.

GelConvertsMired ShiftTransmission
Rosco 3202 Full Blue (CTB)3200K → 5500K−13136% (−1.5 stops)
Rosco 3204 Half Blue (CTB)3200K → 4100K−6852% (−0.9 stop)
LEE 201 Full C.T. Blue3200K → 5700K−137about 35%
Rosco 3407 Full CTO6500K → 3200K+16747% (−1.1 stops)
Rosco 3408 Half CTO5500K → 3800K+8173% (−0.5 stop)
LEE 204 Full C.T. Orange6500K → 3200K+159about 55–63%
Rosco 3405 Sun 85 + ND .35500K → 3200K, minus 1 stop+13133% (−1.6 stops)
Light Falloff (Inverse Square Law)
Illuminance ∝ 1 ÷ Distance²

For a small source, doubling the distance quarters the light (two stops). Moving a light √2 ≈ 1.4 times farther away costs one stop. Large soft sources fall off more slowly up close.

Where

Distance
Distance from the light source(meters)

Example: The same light at 2m and at 4m

Calculation:
(2 ÷ 4)² = 0.25
Result:
A quarter of the light at twice the distance: 2 stops less

Case study

Mixed Lighting: Window Daylight and a Tungsten Key

  • Commercial
  • Advanced
  • Worked example

Scenario

An interior for a commercial: daylight through a window behind the talent, tungsten units for the key, and the camera balanced at 3200K to keep the warm practicals in the shot natural.

Challenge

Bring the window to tungsten color and to an exposure just above the key, which reads at T2.8.

Solution

  1. Read the window and the key with a color meter and an incident meter
  2. Work out the mired shift that turns the window into 3200K
  3. Pick a gel close to that shift, and count the stops it costs
  4. Make up the rest of the exposure difference with ND on the window
  5. Set the key distance with the inverse square law

The numbers

  • Window: 5600K = 178.6 mireds; target 3200K = 312.5 mireds; shift needed +133.9, about +134
  • Rosco 3405 Sun 85 + ND .3 (+131 mireds) brings the window to 1,000,000 ÷ 309.6 ≈ 3230K
  • Window reads f/11 against a T2.8 key: 2 × log₂(11 ÷ 2.8) = 3.9 stops too bright
  • 3405 removes 1.6 stops; add ND 0.6 (2 stops) for 3.6 in total, leaving the window about a third of a stop over the key
  • Key reads T4 at 2m; to read T2.8, move it to 2 × √2 = 2.8m (one stop less)

Result

With these numbers the window lands near 3200K and about a third of a stop above the key, which reads as natural daylight through glass.

Key takeaways

  • Work in mireds: gel values add and subtract directly
  • Count the stops a correction gel costs before choosing ND
  • Moving a light 1.4 times farther away costs one stop
  • Meter and test: real sources are rarely exactly 3200K or 5600K

Calculators for this guide