Mirror Equation Calculator
Solve the spherical mirror equation for image distance and magnification.
What it does
A curved mirror focuses light the way a lens does, but by reflection. Rays from one object point meet again at one image point, and the meeting place follows a strict relation between object distance, image distance, and focal length. That relation predicts where to put the screen and how large the picture will be.
The tool above solves it at once. Choose centimeters or inches, enter object distance and signed focal length, and read image distance, magnification, and type. Defaults place a 30 cm object before a 10 cm concave mirror, so the page opens solved at 15.00 cm and -0.5 x: real and inverted at half size.
Signs carry the meaning. Positive image distance sits in front of the mirror where a card can catch it; negative sits behind where only the eye finds it. Negative magnification flips the arrow; positive keeps it upright. Convex mirrors take negative f and always return a small virtual upright image, which is why car mirrors shrink the view.
The example repays a slow read: 1/10 minus 1/30 leaves 1/15, so the image forms at 15 cm with m = -0.5. Imperial inputs mirror the same geometry (11.81 in, 3.94 in) and return 5.91 in at -0.501 x, the tiny shift coming from rounded inch defaults. Both modes agree on type: real and inverted.
Use the table for classic object spots, then run your layout. The page assumes a spherical mirror with paraxial rays; wide angle beams add aberration that the thin equation cannot show. Within range it answers setup questions fast: screen position, image size, and whether to expect a flip.
How to use
- Pick metric (cm) or imperial (in).
- Enter the object distance.
- Enter the focal length with sign.
- Read image distance, magnification, and type.
- Move the object inside f to watch the image turn virtual.
- Sketch rays to confirm real images before placing a screen.
Formula
- do
- Object distance in front of the mirror.
- di
- Image distance; positive is real, negative is virtual.
- f
- Focal length; positive concave, negative convex.
- m
- Image size and orientation factor.
Distances use one unit per run (cm metric, in imperial). Positive f models a concave mirror and negative f a convex one. When object distance equals focal length the denominator vanishes and the image forms at infinity, which the tool reports as a message rather than a number.
Worked example
Example: object at 30 cm, f at 10 cm
An object stands 30 cm in front of a concave mirror with f = 10 cm. Find the image.
- List inputs: object 30 cm, focal length 10 cm (concave).
- 1/di = 1/10 - 1/30 = 3/30 - 1/30 = 2/30 = 1/15.
- di = 15.00 cm.
- m = -15/30 = -0.5 x.
- Positive di means real; negative m means inverted.
- Imperial mirror: 11.81 in with 3.94 in focus gives 5.91 in at -0.501 x.
Reference table
| Object | Image | Magnification | Type |
|---|---|---|---|
| 40 cm | 13.33 cm | -0.333 x | Real and inverted |
| 30 cm | 15.00 cm | -0.5 x | Real and inverted |
| 20 cm | 20.00 cm | -1 x | Real and inverted |
| 15 cm | 30.00 cm | -2 x | Real and inverted |
| 5 cm | -10.00 cm | 2 x | Virtual and upright |
Limits
- Paraxial rays only; wide angles add aberration.
- Spherical mirrors; aspheric shapes differ.
- Thin equation; thick setups need ray tracing.
- No brightness or sharpness prediction.
Common mistakes
- Dropping the sign on convex focal length.
- Entering zero focal length.
- Reading |m| without the orientation sign.
- Mixing cm and inches in one run.
- Expecting a screen image from a virtual result.
Try it yourself
Quick checks. Answers sit tucked under each question.
Object at 20 cm with f = 10 cm?
di = 20.00 cm. m = -1 x. Real and inverted, same size.
Convex check: object 30 cm with f = -10 cm?
di = -7.50 cm. m = 0.25 x. Virtual and upright.
Object at 40 cm with f = 10 cm? Work it by hand.
1/di = 1/10 - 1/40 = 3/40, so di = 13.33 cm and m = -0.333 x.
Questions
What is the mirror equation?
1/f = 1/do + 1/di, with magnification m = -di/do. Positive f here means a concave mirror; negative f means convex.
Real versus virtual image?
Positive di lands in front of the mirror and can hit a screen. Negative di sits behind the mirror and only the eye sees it.
What does the sign of m say?
Negative m flips top to bottom (inverted). Positive m keeps it upright. Size follows |m|: 0.5 is half size, 2 is double.
Object at the focal point?
At do = f the rays leave parallel, so 1/di = 0 and the image sits at infinity. The tool reports this instead of a number.
What changes for a convex mirror?
A diverging convex mirror with negative f. The same equation gives a virtual upright image behind the mirror.
How do I work in inches?
Pick imperial and enter inches: 11.81 in object with 3.94 in focus gives 5.91 in at about -0.501 x.