Mirror Equation Calculator

Background.

A curved mirror moves images around. Stand well back from a concave mirror and you see yourself small and upside down; step in close and you suddenly appear upright and enlarged, the view a shaving or make-up mirror gives. A convex mirror, like the security mirror at a shop's blind corner, always shows a smaller, upright picture of a wide area. One equation predicts all of this.

The mirror equation, 1/f = 1/d_o + 1/d_i, links three distances measured from the mirror along its axis: the focal length f, the object distance d_o and the image distance d_i. Rearranged as d_i = f·d_o ÷ (d_o − f), it says where the image forms, and the sign of the answer says what kind of image it is.

Because the signs carry the physics, the convention has to be held firmly. Here distances in front of the mirror, on the side where the light actually travels, count as positive. A concave mirror has a positive focal length, half its radius of curvature; a convex mirror has a negative one. A positive image distance means a real image in front of the mirror that a card or screen could catch; a negative one means a virtual image that only appears to lie behind the glass. Mix this with a textbook that uses the opposite convention and every answer flips.

The equation is paraxial: it assumes rays travel close to the axis and meet the mirror at shallow angles. In a wide spherical mirror, rays striking near the rim focus at a slightly different point and blur the image, which is called spherical aberration and is why telescope mirrors are usually ground to a paraboloid or another non-spherical shape. An object placed exactly at the focal point is refused, because the reflected rays leave parallel and the image lies at infinity.

What is mirror equation calculator?

The mirror equation relates where an object sits in front of a curved mirror to where its image forms: 1/f = 1/d_o + 1/d_i. For a spherical mirror the focal length is half the radius of curvature, positive for a concave mirror and negative for a convex one under the real-is-positive convention used here. The page solves for the image distance d_i in metres, and its sign shows whether the image is real (positive, in front of the mirror) or virtual (negative, behind it).

How to use this calculator.

1. 01Enter the focal length in metres, positive for a concave mirror and negative for a convex one. If you know the radius of curvature instead, halve it: a 0.2 m radius gives f = 0.1 m.
2. 02Enter the object distance in metres, measured from the centre of the mirror's surface along its axis and positive for an ordinary object in front of it. Convert centimetres first, so 30 cm becomes 0.3.
3. 03Read the image distance. Positive means a real, inverted image in front of the mirror; negative means a virtual, upright image behind it.
4. 04For the image's size and orientation, work out the magnification m = −d_i ÷ d_o yourself. A negative m means inverted, and a size below 1 means smaller than the object.

The formula.

1/f = 1/d_o + 1/d_i, solved for the image: d_i = f × d_o ÷ (d_o − f). The relation comes from the geometry of reflection at a sphere for rays near the axis, where the focal point lies halfway between the mirror and its centre of curvature. The denominator does most of the work. When the object is far away, d_o − f is close to d_o and the image settles just beyond the focal point. As the object closes on the focal point, the denominator shrinks toward zero and the image races away: with f = 0.1 m, an object at 0.11 m images 1.1 m out. Move the object inside the focal point and the denominator turns negative, putting the image behind the mirror. At d_o = f exactly the image is at infinity, so that input is refused, as are a zero focal length and a zero object distance.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 3, section 2.2, Spherical Mirrors.
2. [2]Hecht, Optics, 5th ed. (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.