Mirror Equation Calculator
Solve the mirror equation 1/f = 1/do + 1/di for whichever value you are missing: image distance, focal length, or object distance. Pick the mirror type, switch units, and get the magnification, radius of curvature, and (if you enter an object height) the image height, all with the working shown.
Formula
Worked example
A concave mirror with f = 10 cm and an object at do = 30 cm: di = (f·do)/(do - f) = (10 × 30)/(30 - 10) = 300/20 = 15 cm. Magnification m = -di/do = -15/30 = -0.5, so the image is real, inverted, and half size. The areal magnification is 0.25 and the radius of curvature is R = 2f = 20 cm.
How the mirror equation works
The mirror equation, 1/f = 1/do + 1/di, ties together the focal length f, the object distance do, and the image distance di for a spherical mirror. This calculator can solve it for any one of those three values: choose what you are missing, enter the other two, and it rearranges the equation for you. Solving for the image gives di = (f·do)/(do - f); solving for the object gives do = (f·di)/(di - f); and solving for the focal length is the direct form f = 1 / (1/do + 1/di). The same relationship is the thin-lens equation, so every result applies equally to converging and diverging lenses. The focal length is half the radius of curvature, so the calculator also reports R = 2f.
Reading the signs and the mirror type
Signs carry the physics, so they matter as much as the numbers. Picking concave (or a converging lens) gives a positive focal length, while picking convex (or a diverging lens) makes it negative; you enter the magnitude and the type selector applies the sign. A positive image distance means the image forms on the same side as the object, in front of the mirror, and is a real image that can be caught on a screen. A negative image distance means the image is virtual and appears to sit behind the mirror, like your reflection in a flat or convex bathroom mirror. The linear magnification m = -di/do encodes both size and orientation: its absolute value is the size ratio, a negative sign means the image is inverted, and a positive sign means it is upright.
Magnification, image height, and areal magnification
Linear magnification compares the image height to the object height. If you enter an object height, the calculator multiplies it by the magnification to give the image height, keeping the sign so you can see the orientation. Areal magnification, the ratio of the image area to the object area, is the square of the linear magnification (m²), which matters when you care about how much light or how large a projected area the image covers. Because magnification squares, an image at twice the size covers four times the area, while a half-size image covers only a quarter of the area.
Special cases to watch for
A few configurations behave unusually. When the object is placed exactly at the focal point (do = f), the denominator do - f becomes zero and the reflected rays emerge parallel, so the image forms at infinity and the calculator returns a blank result rather than a misleading number. For a concave mirror with the object inside the focal length (do < f), the image distance turns negative, producing an upright, magnified virtual image, which is how a shaving or makeup mirror works. A convex mirror always produces an upright, reduced, virtual image regardless of object distance, which is why it gives a wide field of view in car wing mirrors and store security mirrors.
Image type by object position (concave mirror)
| Object position | Image type | Orientation | Size |
|---|---|---|---|
| Beyond C (do > 2f) | Real | Inverted | Reduced |
| At C (do = 2f) | Real | Inverted | Same size |
| Between C and F (f < do < 2f) | Real | Inverted | Enlarged |
| At F (do = f) | No image | , | At infinity |
| Inside F (do < f) | Virtual | Upright | Enlarged |
Behaviour of a concave mirror as the object moves; f is the focal length, C = 2f the centre of curvature.
Frequently asked questions
Can this calculator solve for focal length or object distance, not just image distance?
Yes. Use the "Solve for" selector at the top to choose image distance, focal length, or object distance as the unknown, then enter the other two values. The calculator rearranges 1/f = 1/do + 1/di for whichever variable you picked and shows the working.
What do positive and negative image distances mean?
A positive image distance means a real image forms in front of the mirror on the same side as the object and can be projected onto a screen. A negative image distance means a virtual image appears to sit behind the mirror and cannot be projected, like the reflection in a flat mirror.
What is the difference between linear and areal magnification?
Linear magnification (m = -di/do) is the ratio of image height to object height and carries a sign for orientation. Areal magnification is the ratio of image area to object area and equals the square of the linear magnification (m²), so it is always positive. An image twice as tall covers four times the area.
Does this also work for lenses?
Yes. The mirror equation and the thin-lens equation have the same form, 1/f = 1/do + 1/di, with the same magnification m = -di/do. Choose the converging option for a converging lens (positive focal length) and the diverging option for a diverging lens (negative focal length).
How do I get the image height?
Enter an object height in the optional field and the calculator multiplies it by the magnification to give the image height, keeping the sign so a negative result tells you the image is inverted. Leave the field at zero to skip it.