Thin Lens Calculator - Image, Focal Length and Magnification

The thin lens equation, from a magnifying glass to a projector

Where does the image form, which way up is it, and how big? A lens with a focal length of 15 cm and an object 20 cm away puts a real image 60.00 cm behind the lens, upside down and three times larger. This calculator solves the thin lens equation for any one of its three distances, in millimeters, centimeters or inches, and adds the magnification, the lens power in diopters and a table of where the same lens would put the image for other object positions.

Quick start. Testing an unknown lens on a bench? Choose "Focal length", centimeters, then type the object distance 30 and the distance at which the image is sharp on a card, 60. The lens has a focal length of 20.00 cm, which is +5.00 D.

Where a 10 cm lens puts the image

One converging lens, the object moved closer step by step. The image walks away from the lens and grows, disappears to infinity at the focal point, then comes back on the object's side as a virtual image.

Object distance	Image distance	Magnification	Image
50 cm	12.50 cm	-0.25	real, inverted, smaller
30 cm	15.00 cm	-0.50	real, inverted, smaller
20 cm	20.00 cm	-1.00	real, inverted, same size
15 cm	30.00 cm	-2.00	real, inverted, larger
12.5 cm	50.00 cm	-4.00	real, inverted, larger
10 cm	infinity	none	parallel beam, no image
7.5 cm	-30.00 cm	4.00	virtual, upright, larger
5 cm	-10.00 cm	2.00	virtual, upright, larger

Setting up a lens problem, box by box

Find - the unknown: image distance, object distance or focal length. The other two go in below.
Distances in - millimeters, centimeters or inches. Everything you type and everything the result shows uses that one unit.
Focal length - positive for a converging lens, negative for a diverging one. A lens of +2.50 D has a focal length of 40 cm.
Object distance - from the object to the center of the lens, always above zero.
Image distance - positive for a real image on the far side, negative for a virtual image on the object's side.
Object height (optional) - adds the height of the image, upright or upside down.

Eight lenses doing eight different jobs

1. A textbook converging lens

Situation: focal length 15 cm, object 20 cm away.
Result: image at 60.00 cm, magnification -3.00: real, inverted, three times larger.

The object sits between the focal length and twice it, the zone every projector uses.

2. A magnifying glass

Situation: focal length 10 cm, a stamp 7 cm below the lens.
Result: image at -23.33 cm, magnification 3.33: virtual and upright.

The minus sign is why a magnifier's image cannot be caught on paper: it sits on the stamp's side of the lens.

3. A 50 mm camera lens and a person 2 m away

Situation: focal length 50 mm, object distance 2,000 mm, person 1,700 mm tall.
Result: image at 51.28 mm, magnification -0.0256, image height 43.59 mm.

A full-frame sensor measures 36 by 24 mm, so a 43.59 mm figure does not fit even with the camera turned upright: step back or choose a shorter lens.

4. How far that lens focuses when it moves 2 mm

Situation: find the object distance, focal length 50 mm, sensor 52 mm behind the lens.
Result: the lens is sharp at 1,300.00 mm, magnification -0.0400.

At infinity the sensor would sit exactly 50 mm away, so focusing from far to 1.3 m moves the glass just 2 mm.

5. A slide projector

Situation: focal length 10 cm, a 3.6 cm slide 12 cm from the lens.
Result: sharp on a screen 60.00 cm away, magnification -5.00, image 18.00 cm tall.

The image is inverted, which is why slides go into a projector upside down.

6. A +2.50 D reading lens over a page

Situation: inches, focal length 15.75 in, text 12 in from the lens.
Result: virtual image at -50.40 in, magnification 4.20.

The lens moves the text optically much farther away, where an eye that no longer focuses up close can see it.

7. A -2.00 D lens for nearsightedness

Situation: focal length -50 cm, a sign 500 cm away.
Result: virtual image at -45.45 cm, magnification 0.0909: upright and smaller.

A diverging lens brings distant things optically closer, inside the range a nearsighted eye can focus.

8. Life-size macro at twice the focal length

Situation: millimeters, focal length 100 mm, a coin 200 mm away.
Result: image at 200.00 mm, magnification -1.00.

This is the 1:1 reproduction ratio of macro photography, reached when object and image are both at twice the focal length.

Diopters and focal length side by side

Lens power in diopters is 1 divided by the focal length in meters. Glasses prescriptions use diopters, camera lenses use millimeters, and the table converts between them.

Power	Focal length (cm)	Focal length (in)	Typical lens
+1.00 D	100	39.37	weak reading glasses
+2.50 D	40	15.75	strong reading glasses
+5.00 D	20	7.87	a lens tested on a bench
+10.00 D	10	3.94	a hand magnifier
+20.00 D	5	1.97	a 50 mm camera lens
-2.00 D	-50	-19.69	glasses for mild nearsightedness
-4.00 D	-25	-9.84	glasses for stronger nearsightedness

What people ask about lenses and images

What does a negative focal length mean?

A diverging lens, thinner in the middle than at the edges. It spreads parallel light as if it came from a point in front of the lens. Glasses for nearsightedness are diverging: -50 cm is -2.00 D.

Why did I get a negative image distance?

The image is virtual and sits on the same side as the object. You can see it through the lens but cannot project it. A magnifier with a 10 cm focal length held 7 cm from a stamp gives -23.33 cm.

How do I turn diopters into a focal length?

Divide 1 by the diopters to get meters. +2.50 D is 0.4 m, which is 40 cm or 15.75 in; -4.00 D is -25 cm. Type the result with its sign into the focal length box.

What happens when the object is exactly at the focal point?

The light leaves the lens as a parallel beam and the image forms at infinity, so the result shows no image distance and no magnification. A spotlight works this way. The reverse also holds: a sensor exactly one focal length behind the lens is focused on objects infinitely far away.

Why is the image in a camera upside down?

Almost everything you photograph is farther away than twice the focal length, and there a converging lens always gives an inverted real image. The 50 mm lens and a person at 2 m give a magnification of -0.0256; the minus sign is the flip, which the camera undoes before you see the picture.

Does this work for thick lenses, zoom lenses and mirrors?

Only approximately for thick and multi-element lenses, because their distances are measured from internal reference planes rather than the center of the glass. A curved mirror follows the same equation with the same sign convention, with a focal length of half its radius of curvature.

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