Mirror Equation Calculator - Image Distance and Magnification
Mirror equation calculator that finds image distance, radius of curvature, and linear or areal magnification for concave, convex, and plane mirrors.
Mirror Equation Calculator
Results
What Is the Mirror Equation Calculator?
The mirror equation calculator solves 1/f = 1/v + 1/u for concave, convex, and plane mirrors, returning the image distance, radius of curvature, linear magnification, and areal magnification from the object distance and focal length.
- • Physics homework: Check concave and convex mirror problems that give object distance and focal length and ask for the image position and magnification.
- • Optics lab planning: Predict where a screen should be placed to catch a real, inverted image, or where a virtual image will appear behind a rear-view mirror.
- • Ray diagram checks: Confirm the image position predicted by a ray diagram against the mirror equation result.
- • Sign convention practice: Compare results across concave, convex, and plane mirrors to see how the Cartesian sign convention assigns a sign to each distance.
Most mirror problems in high-school and introductory-college physics give two distances and ask for a third. The mirror equation calculator removes the algebra so the result can be read in seconds.
The default view loads a concave mirror with a 6 cm object distance and a 12 cm focal length. Those values give a virtual, upright, magnified image 12 cm behind the mirror, which is the makeup-mirror case. Switching to a convex mirror with the same numbers produces a virtual, upright, reduced image 4 cm behind the mirror, the car-side-mirror case.
Light is an electromagnetic wave and the mirror equation is the geometric side of that wave behavior, so the Wave Speed Calculator is a natural companion for frequency and wavelength sanity checks on the same optical setup.
How the Mirror Equation Calculator Works
The calculator applies the Gaussian mirror equation 1/f = 1/v + 1/u with the Cartesian sign convention, then uses the linear magnification m = -v/u and the areal magnification m_a = (v/u)^2 to characterise the image.
- u (object distance): Signed distance from the pole to the object. u is entered as a magnitude and recorded as a negative value.
- v (image distance): Signed distance from the pole to the image. Positive v is virtual (behind the mirror); negative v is real (in front).
- f (focal length): Negative for concave mirrors, positive for convex mirrors, infinite for plane mirrors.
- r (radius of curvature): Twice the focal length. The calculator reports r = 2f for spherical mirrors and Infinity for a plane mirror.
The sign convention is applied automatically. The mirror-type selector assigns f = -|f| for concave and f = +|f| for convex; the calculator negates the object distance to enforce u < 0 for a real object.
Solving for v gives v = f*u / (u - f), valid for concave, convex, and the plane-mirror limit when f is taken to infinity. The linear magnification m = -v/u and the areal magnification m_a = (v/u)^2 follow directly. A convex mirror's reduced image gives an areal magnification well below 1, which is why a car-side mirror warns that objects are closer than they appear.
Worked example: concave makeup mirror
Mirror type: concave. Object distance: 6 cm. Focal length: 12 cm (so f = -12 cm).
1/f = 1/v + 1/u -> 1/-12 = 1/v + 1/-6 -> 1/v = 1/12 -> v = +12 cm.
Image distance +12 cm (virtual, behind the mirror), radius -24 cm, linear magnification +2, areal magnification +4.
The image is upright, magnified, and sits 12 cm behind the mirror. That is the classic makeup-mirror geometry: an object between the mirror and its focus produces a magnified virtual image.
According to OpenStax University Physics Vol 3, Section 2.2: Spherical Mirrors, the mirror equation 1/do + 1/di = 1/f relates the object and image distances to the focal length, the linear magnification m = -di/do relates image and object heights, and a plane mirror is the limiting case where the radius of curvature is infinite so do = -di.
According to OpenStax Physics, Section 16.1: Reflection, the radius of curvature of a curved mirror is twice its focal length, and the lens/mirror equation 1/f = 1/di + 1/do gives the same relationship for spherical mirrors as for thin lenses.
The same spherical-mirror geometry that drives this calculator also decides how large a vanity mirror should be, so the Bathroom Mirror Size Calculator applies these optical relationships to a real-world mirror selection.
Key Concepts Explained
The mirror equation looks small, but every variable carries a sign and a physical meaning. The four cards below summarise the quantities the calculator reports and how the Cartesian sign convention interprets them.
Object distance u
Distance from the mirror's pole to the object, measured along the principal axis. In the Cartesian convention, the object is always in front of the mirror, so u is negative; the calculator handles that sign from a positive magnitude.
Image distance v
Distance from the pole to the image. Negative v means a real image on the same side as the object; positive v means a virtual image behind the mirror.
Focal length f
Distance from the pole to the focus where parallel rays converge or appear to diverge. Concave mirrors focus real rays (f negative). Convex mirrors diverge real rays (f positive). Plane mirrors have f at infinity.
Linear magnification m
Ratio of image height to object height, defined as m = -v / u. Positive m means upright, negative m means inverted, |m| > 1 means magnified. Areal magnification is the square of m.
Given any two of u, v, and f, the third is fixed by the mirror equation. Once v is known, m and m_a follow directly. The calculator keeps that chain visible by listing radius (2f) alongside magnification.
The mirror equation shares its image-distance and viewing-angle geometry with screen-viewing math, so the TV Viewing Distance Calculator applies the same focal-length and viewing-angle relationships to a living-room display.
How to Use This Calculator
Pick the mirror type, enter the object distance, and enter the focal length. The result panel updates as you type.
- 1 Choose a mirror type: Use the dropdown to select concave, convex, or plane. The choice sets the sign of the focal length automatically.
- 2 Enter the object distance: Type the magnitude of the object distance in centimeters. The object must sit in front of the mirror, so the value should be positive.
- 3 Enter the focal length: Type the magnitude of the focal length in centimeters. For a plane mirror, this value is ignored because the focal length is infinite.
- 4 Read the result panel: The image distance, radius of curvature, and magnifications update immediately. A positive image distance means a virtual image behind the mirror.
- 5 Reset for a fresh example: Press Reset to return to the default concave mirror with a 6 cm object distance and 12 cm focal length.
A common physics problem asks for the image position when an object sits 6 cm in front of a concave mirror with a focal length of 12 cm. Set the mirror type to Concave, enter 6 cm for object distance, enter 12 cm for focal length, and read the image distance of 12 cm. The positive sign means a virtual, upright, magnified image 12 cm behind the mirror, which is the makeup-mirror case.
The mirror equation 1/f = 1/v + 1/u has the same algebraic form as Hooke's law for springs, so the Spring Constant & Deflection Calculator applies that parallel F = kx relationship to mechanical deflection problems.
Benefits and When to Use It
The mirror equation calculator turns a three-variable algebra problem into a single result panel. Specific reasons to reach for this tool include:
- • Faster homework checks: Skip the algebra and check the final image distance, magnification, and image type for concave, convex, and plane problems.
- • Sign convention handled automatically: The calculator applies the Cartesian sign convention to u and f based on the mirror type, removing the most common source of mirror-equation mistakes.
- • Side-by-side mirror comparisons: Switching the mirror type dropdown with the same object distance and focal length makes it easy to see why a convex passenger-side mirror warns that objects are closer than they appear.
- • Magnification and radius in one place: Linear and areal magnification plus the radius of curvature appear together, the complete set of mirror outputs needed for most physics problems.
- • Plan a real lab setup: Predict where a screen should sit to catch a real, inverted image from a concave mirror before any hardware is moved.
The calculator is best for spherical mirror problems where the paraxial approximation holds. For a single quick check, the default concave example already covers the makeup-mirror case.
Optical mounts, telescope tubes, and laser housings sit on the same simple-harmonic-motion model that the mirror equation shares with mechanical oscillators, so the Vibration Natural Frequency Calculator covers the natural frequency of those support structures.
Factors That Affect Results
The mirror equation is exact for spherical mirrors under the paraxial approximation, but four factors determine how closely the calculated image matches the real image in front of you.
Sign convention
The mirror equation uses signed distances. Mixing conventions across the same problem is the most common source of sign errors, so the calculator applies the Cartesian convention consistently and exposes the sign in the result panel.
Mirror type
Concave mirrors can produce either real, inverted images or virtual, upright images depending on object position. Convex mirrors always produce virtual, upright, reduced images.
Object position
An object at the focus sends v to infinity and breaks the linear magnification. An object at the center of curvature gives v = u and m = -1. Both are useful sanity checks because the result is well known from ray diagrams.
Radius of curvature
A larger radius means a longer focal length, a larger object distance before v goes to infinity, and a larger radius row in the result panel. The radius is twice the focal length for any spherical mirror.
- • The mirror equation is a paraxial approximation. Rays that strike the mirror far from the principal axis do not all meet at a single point, and the result drifts from the real image as the aperture grows.
- • Spherical mirrors also suffer from spherical aberration, which is why telescopes and headlights use parabolic mirrors for high-quality imaging.
For the cleanest results, use the same distance units (centimeters by default) for every entry, and confirm the sign of v by reading whether the image is in front of or behind the mirror before comparing to a ray diagram.
According to Physics LibreTexts, Section 2.3: Spherical Mirrors, the small-angle (paraxial) approximation is what makes 1/do + 1/di = 1/f a valid result, and the sign convention is positive f for concave mirrors and negative f for convex mirrors with positive di for real images and negative di for virtual ones.
A parabolic solar collector shows how a curved mirror concentrates light into a useful beam, so the Work, Energy & Power Calculator covers the power-per-area ratio that controls how much useful energy a mirror can deliver per unit of incident radiation.
Frequently Asked Questions
Q: What is the mirror equation formula?
A: The mirror equation is 1/f = 1/v + 1/u, where f is the focal length, v is the image distance, and u is the object distance. The Cartesian sign convention assigns a sign to each variable: u is negative when the object is in front of the mirror, f is negative for concave mirrors and positive for convex mirrors, and v is positive for virtual images behind the mirror.
Q: How do you find the image distance using the mirror equation?
A: Rearrange 1/f = 1/v + 1/u to solve for v. The result is v = f*u / (u - f). Plug in the signed object distance u and the signed focal length f, both of which the calculator fills in from the chosen mirror type, and the image distance appears in the result panel.
Q: What is the difference between linear and areal magnification?
A: Linear magnification m = -v / u compares the image height to the object height. Areal magnification m_a = (v / u)^2 compares the image area to the object area. The areal value is the square of the linear value, so a linear magnification of 0.5 means the image covers 0.25 of the object's area.
Q: Why is the focal length of a plane mirror infinity?
A: A plane mirror neither converges nor diverges light. Parallel rays stay parallel after reflection, so they never meet at a focus. Mathematically, f is taken to infinity, and the mirror equation reduces to v = -u, which means the image is at the same distance behind the mirror as the object is in front.
Q: What is the Cartesian sign convention for mirrors?
A: In the Cartesian convention, distances are measured from the pole of the mirror. The object is in front, so u is negative. Concave mirrors focus real rays, so f is negative. Convex mirrors diverge real rays, so f is positive. Real images sit in front of the mirror and carry a negative v, while virtual images sit behind the mirror and carry a positive v.
Q: How are the radius of curvature and focal length related?
A: For a spherical mirror, the focal length is half the radius of curvature, so r = 2f. The calculator reports both rows so the relationship stays visible. A plane mirror has an infinite radius of curvature because its surface is not part of a sphere, which is why its focal length is taken as infinity.