Smartphone Projector Calculator (DIY box: lens focal length → box length and picture size)

Understanding the smartphone projector calculator — concept and principle

This calculator designs a do-it-yourself shoebox projector: from the lens focal length, the phone's screen size and the distance to the wall, it gives the distance between phone and lens, the magnification, the picture size and diagonal on the wall and how bright the picture is compared with the screen. It is a popular science project and a good thin-lens exercise.

Place the phone just beyond the focal length; the image grows 12 times but spreads its light over 150 times the area.

How it is calculated

Thin-lens equation: 1/d_o + 1/d_i = 1/f, so d_o = 1 ÷ (1/f − 1/d_i)

Magnification m = d_i ÷ d_o

Picture width and height = screen size × m; diagonal in inches = √(w² + h²) ÷ 2.54

Brightness relative to the screen ≈ 1 ÷ m²

The principle

A converging lens forms a real, enlarged image of the phone screen on the wall when the screen sits just beyond the focal length. The further the wall, the closer the screen must be to f and the bigger the picture. The same light is spread over m² times the area, so the picture is much dimmer than the screen. The image is upside down and mirrored, which is why the phone is placed upside down with rotation locked.

Worked example

Default inputs: f = 150 mm, screen 140 × 65 mm, wall at 2 m.

d_o = 1 ÷ (1/150 − 1/2,000) = 162 mm from screen to lens.
m = 2,000 ÷ 162.16 = 12.33×.
Picture = 140 × 12.33 = 173 cm wide and 65 × 12.33 = 80 cm high, a 75-inch diagonal.
Brightness ≈ 1 ÷ 12.33² = 0.006574 of the screen's.

Each point is the calculator re-run with only “Lens focal length f (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Lens focal length f (mm)	Phone screen to lens distance (mm)
37.5	38
75	78
112	119
150	162
188	207
225	254
300	353

How to read the result

A 75-inch picture is large, but at well under 1% of the screen's brightness it needs a dark room and full screen brightness. A lens with a larger diameter collects more light. Moving the wall closer makes a smaller but brighter picture.

Common mistakes

Placing the phone at exactly the focal length, which sends the image to infinity.
Forgetting the image is inverted and mirrored.
Using a lens with a small diameter, which gives a very dim picture.

Keep in mind: The estimate ignores lens aberrations and light lost inside the box. Do not leave a phone at full brightness in a closed box for long, as it can overheat.

Frequently asked questions

Why must the phone be upside down?

A single lens inverts the image.

How do I make the picture bigger?

Move the wall further away or use a shorter focal length.

Why is the picture so dim?

The light is spread over an area m² times larger than the screen.

What lens works best?

A large-diameter magnifying lens, such as a Fresnel lens, with a focal length suited to your box.

Sources: OpenStax, University Physics Volume 3 (2016), section 2.4 "Thin Lenses"

Checked 9 Nov 2026. Thin-lens equation 1/d_o + 1/d_i = 1/f and magnification m = −d_i/d_o: OpenStax, University Physics Volume 3 (2016), Chapter 2 Key Equations (§2.4 “Thin Lenses”).

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