Mirror equation calculator
Enter any two of f, d_o, d_i and m, get the rest and the type of image.
Fill in any two of the four quantities below; leave the others blank. Signs: f > 0 concave, f < 0 convex; d_i > 0 real image, d_i < 0 virtual.
What is the mirror equation?
For a spherical mirror, 1/d_o + 1/d_i = 1/f, where d_o is the object distance, d_i the image distance and f the focal length, which is half the radius of curvature, f = R/2. The magnification is m = h_i/h_o = −d_i/d_o (OpenStax University Physics, Volume 3, section 2.3). The equation holds for rays close to the axis (the small-angle approximation). Fill in any two of the four quantities and the calculator finds the rest; the pre-filled example is a concave makeup mirror with f = 12 in and your face 6 in away.
What do the signs mean?
The sign convention is the one OpenStax uses: f is positive for a concave mirror and negative for a convex one (so R is negative for convex); d_i is positive for a real image, which forms in front of the mirror and can be projected on a screen, and negative for a virtual image behind the mirror. The object is real, so d_o is always positive. A positive magnification means an upright image, a negative one an inverted image; |m| above 1 means enlarged and below 1 reduced. Lenses follow the same rule for f and d_i, so this matches the thin lens calculator.
Example: what focal length makes a makeup mirror magnify 1.5 times?
OpenStax problem 35: a face 12.0 cm from the mirror, magnification 1.50. The image distance is d_i = −m × d_o = −18.0 cm, negative, so the image is virtual (behind the mirror), and 1/f = 1/12.0 + 1/(−18.0) gives f = +36.0 cm: positive, so the mirror is concave. Enter the object distance and the magnification, leave the rest blank, and the calculator returns the same f, R = 72 cm and d_i = −18 cm.
How do I use a mirror with the Sun or a distant object?
Choose “Very far away” for the object: its distance is then infinite, 1/d_o = 0, and the image forms at the focal point, d_i = f. OpenStax Example 2.1: to focus sunlight 40.0 cm from a solar trough mirror, f = 40.0 cm, so the radius is R = 2f = 80.0 cm. If instead the object sits exactly at the focal point (d_o = f), the reflected rays leave parallel and no image forms, which is how a searchlight or a flashlight reflector works.
Example: what is the focal length of a convex mirror?
OpenStax Example 2.2: the cornea acts as a convex mirror; a light 12 cm away makes an image of magnification 0.032. Then d_i = −0.032 × 12 = −0.384 cm and f = (1/12 + 1/(−0.384))⁻¹ = −0.397 cm, so R = 2f = −0.79 cm (the book rounds it to −0.80 cm, and prints the intermediate f as −40.0 cm, a slip). Convex mirrors always give virtual, upright, reduced images of real objects, which is why they are used for security mirrors and car mirrors.
Sources: OpenStax University Physics, Volume 3, section 2.3 “Spherical Mirrors” (mirror equation, sign convention, Examples 2.1 and 2.2) and the chapter 2 problems and answer key. The lens version: thin lens calculator; refraction: Snell’s law calculator; related: wavelength calculator and photoelectric effect calculator.