Mirror Equation Calculator

Mirror Equation

For a curved mirror with focal length f, an object at distance d_o produces an image at distance d_i where:

1/f = 1/d_o + 1/d_i

The magnification (size ratio) is:

m = −d_i / d_o

Sign Convention (Cartesian / “real-is-positive”)

The radius of curvature R is related by R = 2f.

Image Type Quick Reference

Worked Example: Concave Mirror, f = 10 cm, d_o = 30 cm

• 1/d_i = 1/10 − 1/30 = 3/30 − 1/30 = 2/30
• d_i = 15 cm (positive → real image)
• m = −15/30 = −0.5 (inverted, half-size)

So the image forms 15 cm in front of the mirror, upside down, at half the object’s size. That is exactly what a shaving mirror does when you hold it too far from your face.

Worked Example: Convex Car Side Mirror

Convex mirrors always produce virtual, upright, reduced images. That is why side mirrors are stamped “objects in mirror are closer than they appear”. The reason is the shrinking, not the image position: your brain judges distance largely from how big something looks, so a car rendered at half size reads as roughly twice as far away. The mirror is curved to buy a wider field of view, and the warning label is the price of that trade.

Limitations

The thin-mirror equation assumes paraxial rays, meaning rays close to the optical axis. For wide-aperture or high-speed optics, spherical aberration, coma, and astigmatism produce real-world deviations from the simple formula. Telescopes use parabolic instead of spherical mirrors specifically to eliminate spherical aberration on-axis.