Mirror Equation
Solve the mirror equation 1/f = 1/v + 1/u to compute object distance, image distance, focal length, radius of curvature, and magnification of concave, convex, and plane mirrors. Free online physics optics calculator with interactive charts and step-by-step breakdown.
Enter any two values to compute the third using 1/f = 1/v + 1/u. For concave mirrors use f > 0; for convex mirrors use f < 0.
About This Calculator
The Mirror Equation Calculator helps physics students, optics enthusiasts, and engineers analyze curved mirror systems using the fundamental mirror formula 1/f = 1/v + 1/u. By entering any two of the three values — object distance (u), image distance (v), or focal length (f) — the calculator instantly determines the missing parameter, along with radius of curvature, linear and areal magnification, mirror type (concave, convex, or plane), and image characteristics (real or virtual). This tool is ideal for solving optics problems in high school and college physics, laboratory work, telescope and reflector design, and everyday mirror applications.
The mirror equation is derived from geometric optics under the paraxial approximation, where light rays make small angles with the optical axis. For spherical mirrors, the equation 1/f = 1/v + 1/u accurately predicts image formation. The Cartesian sign convention used is: object distance u is always positive for real objects placed in front of the mirror; image distance v is positive for real images (formed in front of the mirror, inverted) and negative for virtual images (formed behind the mirror, upright); focal length f is positive for concave (converging) mirrors and negative for convex (diverging) mirrors. The radius of curvature is related to focal length by r = 2f. Linear magnification is m = −v/u, where a negative value indicates an inverted image. Areal magnification is mₐ = v²/u².
Formula: 1/f = 1/v + 1/u, where u = object distance (cm), v = image distance (cm), f = focal length (cm). r = 2f. Linear magnification m = −v/u. Areal magnification mₐ = v²/u².
How to use: Enter any two values in centimeters (use positive f for concave mirrors, negative f for convex mirrors). Leave the unknown field blank. Click Calculate to solve for all parameters. The results display the computed values, mirror and image type classification, a bar chart comparing the three distances, and a pie chart showing the optical power distribution.
Regional Notes
India (IN): The mirror equation is taught in Class 10, Class 12, and undergraduate physics curricula following CBSE and NCERT standards. Sign conventions for mirrors are covered extensively in the optics chapter, where distances measured against the direction of incident light are taken as negative under the Cartesian sign convention. Students learn image formation through ray diagrams for both concave and convex mirrors.
United States (US): The mirror equation is covered in high school physics, AP Physics 2, and introductory college optics courses. US textbooks commonly use the formula 1/do + 1/di = 1/f where do = object distance and di = image distance. Sign conventions differ by textbook, but the underlying mathematical relationship is identical. The mirror equation is essential for understanding telescopes, microscopes, and optical instruments.
United Kingdom (UK): The mirror equation is part of A-Level Physics and GCSE Physics optics modules. UK curricula emphasize real and virtual image formation through ray diagrams and the mirror formula. Students are assessed on sign convention application and magnification calculations for both concave and convex mirrors in practical optics contexts such as shaving mirrors, rearview mirrors, and reflecting telescopes.
Frequently Asked Questions
What is the mirror equation?
The mirror equation relates the object distance (u), image distance (v), and focal length (f) of a spherical mirror: 1/f = 1/v + 1/u. It is used to determine where an image will form for a given object placed in front of a concave or convex mirror. For a plane mirror, the focal length is considered infinite, giving v = −u.
How do I calculate focal length from object and image distance for a mirror?
Use the mirror equation: 1/f = 1/v + 1/u. If you know the object distance u and image distance v, the focal length f = 1 / (1/u + 1/v). For example, with u = 30 cm and v = 15 cm for a concave mirror, f = 1 / (1/30 + 1/15) = 10 cm. Enter any two values in the calculator to compute the third.
What is the difference between a concave and convex mirror?
A concave mirror (converging) has a positive focal length and brings parallel light rays together at a focal point in front of the mirror. It can form real or virtual images depending on the object position. A convex mirror (diverging) has a negative focal length and spreads parallel light rays outward, always forming virtual, upright, and diminished images regardless of object position.
What does a positive vs negative image distance mean for mirrors?
A positive image distance (v > 0) means the image forms in front of the mirror — this is a real image that can be projected onto a screen and is inverted. A negative image distance (v < 0) means the image forms behind the mirror — this is a virtual image that cannot be projected and is upright, visible only by looking into the mirror.
How do I determine if an image is real or virtual for a mirror?
For a concave mirror: if the object is beyond the focal point (u > f), the image is real and inverted. If the object is between the pole and the focal point (u < f), the image is virtual, upright, and magnified (like a makeup mirror). For a convex mirror: the image is always virtual, upright, and smaller than the object, regardless of the object position.
What is the radius of curvature of a mirror?
The radius of curvature (r) of a spherical mirror is the radius of the sphere of which the mirror is a part. It is related to the focal length by the formula r = 2f. For a concave mirror, both f and r are positive; for a convex mirror, both are negative. The radius of curvature determines how strongly the mirror converges or diverges light.
How are linear and areal magnification calculated for mirrors?
Linear magnification (m) is the ratio of image height to object height, calculated as m = −v / u for mirrors. A negative m indicates an inverted image; a positive m indicates an upright image. Areal magnification (mₐ) is the ratio of image area to object area, calculated as mₐ = v² / u². It is always positive and equals the square of linear magnification.
What happens when the object is placed at the focal point of a concave mirror?
When the object is placed exactly at the focal point of a concave mirror (u = f), the mirror equation gives 1/v = 1/f − 1/f = 0, meaning v → ∞. The reflected rays emerge parallel and no image is formed on a screen — the image is said to be at infinity. This principle is used in flashlights and searchlights to produce parallel beams of light.