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Mirror equation: image distance, magnification and image height

Find where a concave or convex mirror forms the image of an object, how large it is and whether it is real or virtual, upright or inverted.

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Results

Image distance

15.00

Real image, in front of the mirror: it can be projected on a screen

Magnification

-0.500

Inverted and reduced
Image height
-1.00
Radius of curvature
20.00

Images in curved mirrors

A spherical mirror forms an image whose position and size depend on its focal length and on how far the object stands from it. Concave mirrors, which curve inward like the inside of a spoon, can make real images that you can project on a screen, or enlarged virtual images when you stand close. Convex mirrors, which bulge outward, always make small upright images that show a wide field of view.

The formulas

With every distance measured from the mirror and counted positive in front of it:

  • Mirror equation: 1/f = 1/do + 1/di, so di = f × do ÷ (do − f)
  • Magnification: m = −di ÷ do
  • Image height: hi = m × ho
  • Radius of curvature: R = 2f

A concave mirror has a positive focal length and a convex mirror a negative one. The calculator applies the sign for you: you enter the focal length as a positive number and choose the type of mirror.

Example

A concave mirror with a focal length of 10 cm and an object 2 cm tall placed 30 cm away:

  • Image distance: 10 × 30 ÷ (30 − 10) = 15 cm, in front of the mirror, so the image is real
  • Magnification: −15 ÷ 30 = −0.5, so the image is inverted and half size
  • Image height: −1 cm
  • Radius of curvature: 20 cm

Reading the result

  • Positive image distance: a real image in front of the mirror, which can be caught on a screen.
  • Negative image distance: a virtual image behind the mirror, seen only by looking into it.
  • Negative magnification: the image is upside down. Positive: it is upright.
  • |m| greater than 1: the image is larger than the object.

When the object sits exactly at the focal point of a concave mirror, the reflected rays leave parallel and no image forms. That is how headlights, flashlights and solar cookers use concave mirrors, in reverse.

Where curved mirrors are used

Telescopes use large concave mirrors to collect light, dentists and makeup mirrors use them to magnify, and stores, parking garages and passenger-side car mirrors use convex mirrors to cover blind spots. For lenses, which follow a similar equation with different sign rules, use the thin lens calculator.

Frequently asked questions

What is the mirror equation?

1/f = 1/do + 1/di, where f is the focal length, do the distance from the object to the mirror and di the distance from the image. The magnification is m = −di ÷ do.

What sign conventions does the calculator use?

Distances in front of the mirror are positive. A concave mirror has a positive focal length and a convex mirror a negative one. A positive image distance means a real image in front of the mirror; a negative one, a virtual image behind it.

When does a concave mirror make an upright image?

When the object is closer than the focal point. The image is then virtual, upright and enlarged, which is how shaving and makeup mirrors work.

Why are car side mirrors convex?

A convex mirror always gives a virtual, upright and reduced image, so it shows a wider field of view. Objects look smaller and farther away than they are, hence the warning printed on passenger-side mirrors.

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Results are estimates provided for information purposes. They do not replace professional advice.

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