How this instrument works
A shoebox smartphone projector works because a converging lens bends light from the phone's screen so it lands, magnified, on a distant wall. The full physics is the thin-lens equation, 1/do + 1/di = 1/f, but a shoebox build simplifies it: the phone is slid back and forth until the picture is sharp, which happens when the object distance do sits almost exactly at the focal length f. At that setting the image distance di grows large and very nearly equals the throw distance D — the gap from lens to wall — so magnification, normally m = di/do, collapses to the tidy ratio M = D ⁄ f used here.
The formula's shape explains the build's two knobs directly. Magnification rises with throw distance, because pushing the wall farther away stretches the same cone of light over more area, and it rises as the lens gets shorter in focal length, because a stronger curve bends light more sharply per millimetre of lens. The approximation holds well once D is much larger than f — a ratio of ten or more is comfortable, and the 40-to-1 ratio in the worked example below is well inside that range — but it degrades as the box gets short and the throw distance approaches the focal length, where the simplified formula and the real optics part company.
The genuine limit is not sharpness but light. A phone backlight puts out a fixed amount of light no matter how the lens is arranged, and spreading that fixed amount across a bigger picture makes every point on the wall dimmer, roughly in proportion to the square of the magnification. People assembling their first shoebox projector often expect a longer box to simply mean a bigger, equally bright picture; instead it buys a bigger, markedly dimmer one, which is why every guide insists on a blacked-out room rather than a brighter bulb.
- Enter the Phone screen width — measure the visible display edge to edge, not the case around it.
- Enter the Magnifying lens focal length, printed on the lens mount or found by focusing sunlight to a point and measuring that distance.
- Set the Distance from lens to wall/screen to the planned throw — how far the box will sit from the projection surface.
- Read Magnification, × to see how many times larger the projected picture is than the phone screen itself.
- Read Projected image width for the resulting picture size at that throw distance, in the same length unit you entered.
Worked example — a shoebox projector's 2 m throw
A typical shoebox build: a 4-inch-wide phone screen (0.1016 m), a 50 mm magnifying lens (0.05 m focal length) fixed at the box's open end, and the box propped 2 m from a bedroom wall. Magnification comes first: M = D ⁄ f = 2 ⁄ 0.05 = 40. Image width follows directly: w_image = w_screen × M = 0.1016 × 40 = 4.064 m — a wall-filling picture roughly 160 inches across, built from a phone screen small enough to cover with one hand.
That 4.064 m width sounds like a home-cinema screen, and geometrically it is one — but the light powering it still comes from a phone backlight sized to illuminate a display of about 0.007 square metres, now stretched across nearly 6.6 square metres of wall. The picture is genuinely that size and genuinely that dim, which is exactly why commercial pico-projectors abandon a phone's own backlight for a dedicated LED or laser source shining through a real projection lens, rather than a magnifying glass borrowed from a desk drawer.
Questions
Why does magnification equal throw distance divided by focal length, and not the full lens equation?
Because a shoebox projector's phone sits just beyond the lens's focal point — the classic slide-it-until-sharp adjustment. At that setting the thin-lens result 1/do + 1/di = 1/f reduces to di ≈ D and do ≈ f, so magnification m = di/do simplifies to D/f. The approximation is accurate whenever the throw distance is much larger than the focal length, as here: 2 m against 50 mm, a ratio of 40.
Why does the picture get dimmer as the throw distance increases?
Because a fixed amount of light from the phone's backlight gets spread over a larger area as magnification grows. Image area scales with the square of magnification, so doubling the throw distance roughly quarters the brightness. This is why large DIY projections need a fully dark room even though the size itself follows a simple straight-line formula.
Can I use any magnifying glass, or does the focal length matter?
The focal length sets both the magnification and how precisely the phone must be positioned. A shorter focal length gives more magnification at a given throw distance but shrinks the zone where the image stays sharp. A 50-75 mm lens, as used in the worked example, is a common choice because it balances a workable image size against a forgiving focus range inside a shoebox.
What happens if I move the phone instead of the lens or wall?
The image goes out of focus rather than changing size predictably. This calculator assumes the phone sits at the sharp-focus distance from the lens, close to the focal length; sliding the phone changes the object distance, not the throw distance, and steps outside the simplified D/f relationship. Refocus by inching the phone back and forth until the projected image sharpens, then read the throw distance to the wall.
How do I get the projected image height, not just width?
Multiply the projected image width by your phone screen's aspect ratio. A 16:9 screen projecting 4.064 m wide gives roughly 2.29 m tall (4.064 × 9 ⁄ 16); a taller-aspect screen scales proportionally differently. Width is what the calculator reports because phone aspect ratios vary, but the same magnification factor applies uniformly in both directions.
Does the type of lens, glass versus Fresnel, change the formula?
No — magnification here depends only on focal length and throw distance, not on how the lens is manufactured. A Fresnel lens and a glass magnifying lens sharing the same focal length produce the same M and the same image width; what differs is image quality, since a thin Fresnel lens typically shows more distortion and colour fringing at high magnification than solid glass.