Thin lens equation calculator: image distance and magnification

The thin lens equation

1/f = 1/dₒ + 1/dᵢ

• f is the focal length of the lens.
• dₒ is the distance from the object to the lens.
• dᵢ is the distance from the lens to the image.

Usually f and dₒ are known and you want the image. Rearranged: dᵢ = 1 ÷ (1/f − 1/dₒ), or equivalently dᵢ = f·dₒ ÷ (dₒ − f). In Excel: =1/(1/B4-1/C4). For the close-up, 1 ÷ (1/50 − 1/150) = 1 ÷ 0.01333 = 75 mm. All three distances must be in the same unit. Millimetres are convenient for camera lenses.

Worked example: seven lens setups in one table

Focal lengths and object distances are inputs (yellow), in millimetres. Column D solves the thin lens equation for the image distance, column E calculates the magnification and column F names the image type. Rows 10 to 13 size the projected slide, convert the projector lens to dioptres and check the close-up by solving back for f.

Worked example: seven lens setups in one table

	A	B	C	D	E	F
1	Setup	f (mm)	dₒ (mm)	dᵢ (mm)	m	Image
2	Camera, subject at 5 m	50	5,000	50.5	-0.010	Real, inverted
3	Portrait lens at 2 m	85	2,000	88.8	-0.044	Real, inverted
4	Close-up of a flower	50	150	75.0	-0.500	Real, inverted
5	Object at 2f	100	200	200.0	-1.000	Real, inverted
6	Slide projector	100	105	2,100.0	-20.000	Real, inverted
7	Magnifying glass	100	60	-150.0	2.500	Virtual, upright
8	Diverging lens	-200	300	-120.0	0.400	Virtual, upright
9
10	Slide height in the projector (mm)					24
11	Image height on the screen = |m| × slide height (mm)					480
12	Power of the projector lens = 1000 ÷ f (dioptres)					10.0
13	Check: f back from dₒ and dᵢ of the close-up (mm)					50.0

Ideal thin lenses, distances in millimetres, real-is-positive sign convention. Educational calculations only, not eye-care or engineering advice.

Sign convention: real is positive

The calculator uses the common "real is positive" convention:

• f > 0 for a converging (convex) lens, f < 0 for a diverging (concave) lens.
• dᵢ > 0: a real image on the far side of the lens, which you can catch on a screen or sensor.
• dᵢ < 0: a virtual image on the same side as the object, which you only see by looking through the lens.

Some textbooks use the Cartesian convention instead, with the lens formula written 1/v − 1/u = 1/f and object distances counted as negative. The physics is identical, only the signs move. Pick one convention and keep it for the whole problem.

Magnification and image type

The magnification is m = −dᵢ ÷ dₒ. A negative m means the image is upside down, a positive m that it is upright, and |m| > 1 that it is larger than the object. The image height is m times the object height.

For a single lens, a real image is always inverted and a virtual image always upright, so one IF on dᵢ gives the image type: =IF(D2>0,"Real, inverted","Virtual, upright"). The slide projector shows the effect of a large m: a 24 mm slide just outside the focal length of a 100 mm lens appears 2,100 mm away at m = −20, so the picture on the screen is 480 mm high.

Where the image lands: the five cases

• Object far away: the image sits almost exactly at f. A 50 mm camera lens focused on a subject 5 m away is only 50.5 mm from the sensor, half a millimetre further than at infinity.
• Object at 2f: the image is also at 2f, real, inverted and the same size (m = −1).
• Object between f and 2f: a large real image far away, the projector case.
• Object inside f: a virtual, upright, enlarged image. The magnifying glass with the object 60 mm from a 100 mm lens gives dᵢ = −150 mm and m = 2.5.
• Object exactly at f: the rays leave parallel and no image forms. The formula divides by zero, and Excel shows #DIV/0!.

A diverging lens always gives a virtual, upright, smaller image: f = −200 mm with an object at 300 mm gives dᵢ = −120 mm and m = 0.4.

From focal length to dioptres

Opticians describe lenses by their power in dioptres: P = 1 ÷ f with f in metres, or P = 1000 ÷ f with f in millimetres. The 100 mm projector lens has a power of 10 D. Thin lenses in contact simply add their powers, which is why glasses are written in dioptres rather than focal lengths. Spectacle lenses come in 0.25 D steps, so a calculated power is rounded with =ROUND(P*4,0)/4, the same trick as in rounding to the nearest 0.05.

Mirrors use the same equation

Curved mirrors follow the same relationship, 1/f = 1/dₒ + 1/dᵢ, with the focal length equal to half the radius of curvature, f = R ÷ 2. A concave mirror has f > 0 and works like a converging lens, a convex mirror has f < 0 and, like a car's passenger-side mirror, always shows a smaller upright virtual image.

Common mistakes

• Mixed units. A focal length in millimetres and an object distance in metres give nonsense. Convert everything to one unit first.
• Mixed sign conventions. A formula from one textbook with signs from another flips real and virtual images. Use one convention throughout.
• Dropping the minus in m. m = dᵢ ÷ dₒ without the minus sign makes inverted images look upright.
• An object at the focal point. 1/f − 1/dₒ is zero and the formula returns #DIV/0!. Wrap it in IF(C2=B2,"No image",…) if your inputs can hit f.
• Applying it to thick lenses. Real camera lenses contain many elements. The equation still works, but distances are measured from the lens's principal planes, not from the front glass.
• Dioptres from millimetres. P = 1 ÷ f needs f in metres. With f in millimetres use 1000 ÷ f, or a 100 mm lens comes out as 0.01 D instead of 10 D.