Free Mirror Equation Calculator

Understanding the Mirror Equation and Its Calculator

This interactive mirror equation calculator is designed to help you determine any unknown quantity among object distance, image distance, focal length, and radius of curvature. It serves as a concave mirror calculator, convex mirror calculator, and plane mirror calculator, handling all spherical and flat mirror scenarios. The tool also computes both linear magnification (height ratio of image to object) and areal magnification (area ratio), while automatically enforcing the Cartesian sign convention to eliminate common sign mistakes.

The Mirror Formula for Curved Mirrors

The mirror formula relates the object distance $u$, image distance $v$, and focal length $f$ of a spherical mirror. The standard equation is:

$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

where:

• $f$ = focal length (distance from the mirror’s pole to the principal focus),
• $v$ = image distance,
• $u$ = object distance.

This formula also applies to plane mirrors. Because the focal length of a plane mirror is effectively infinite ($f \to \infty$), the term $1/f$ becomes zero, simplifying the equation to:

$$0 = \frac{1}{v} + \frac{1}{u} \quad \Rightarrow \quad v = -u$$

Thus, for a plane mirror, the image lies exactly as far behind the mirror as the object is in front, confirming that the image is always virtual and upright.

Incorporating the Radius of Curvature

Every spherical mirror has a radius of curvature $r$, defined as the radius of the sphere from which the mirror is derived. The focal length is exactly half of this radius:

$$f = \frac{r}{2}$$

Substituting this into the mirror formula yields an equivalent form:

$$\frac{2}{r} = \frac{1}{v} + \frac{1}{u}$$

The calculator automatically switches between these two forms when solving for $f$ or $r$.

Linear and Areal Magnification

Two types of magnification describe how the image compares to the object:

• Linear magnification ($m_{\text{linear}}$) compares image height to object height:

$$m_{\text{linear}} = -\frac{v}{u}$$

The negative sign follows the sign convention: real (inverted) images produce a negative magnification, while virtual (upright) images give a positive value. This formula is valid for all concave and convex mirrors, regardless of whether the image is real or virtual.

• Areal magnification ($M_{\text{areal}}$) compares image area to object area. It equals the square of the linear magnification:

$$M_{\text{areal}} = m_{\text{linear}}^2 = \left( \frac{v}{u} \right)^2$$

Once you enter the required distances, the calculator displays both magnification values.

How to Use the Mirror Equation Calculator

The tool adapts to the three main mirror types and reminds you to follow the Cartesian sign convention.

• Concave mirror mode (default): Enter the object distance as a negative number and the focal length as a negative number. The calculator returns the image distance, radius of curvature, linear magnification, and areal magnification.
• Convex mirror mode: Switch to this mode. Input a negative object distance and a positive focal length. The results include a positive image distance (indicating a virtual image behind the mirror), a positive radius of curvature, and the corresponding magnifications.
• Plane mirror mode: Select this option and enter a negative object distance. The tool sets the focal length and radius of curvature to infinity, displays $v = -u$, and shows both linear and areal magnifications equal to 1.

By following the on‑screen prompts, you avoid sign‑convention errors and obtain accurate results.

Image Locations for Curved Mirrors

Concave Mirrors

The position of the image shifts according to the object’s location relative to the center of curvature (C), the principal focus (F), and the pole (P):

• Object at infinity → Image at F.
• Object beyond C → Image between F and C.
• Object at C → Image also at C.
• Object between C and F → Image beyond C.
• Object at F → Image at infinity.
• Object between P and F → Virtual image behind the mirror.

Convex Mirrors

For a convex mirror, the image is always virtual, erect, and diminished. The exact location depends on the object distance:

• Object at infinity → Image at F behind the mirror.
• Object at any other distance → Image between P and F behind the mirror.

Convex mirrors cannot produce real images. They always diverge incident light, so reflected rays never converge in front of the mirror; they only appear to diverge from a point behind it.

Cartesian Sign Convention Summary

To obtain correct results, keep these sign rules in mind:

• Distances measured from the mirror’s pole: Points in front of the mirror (real objects, real images) are negative; points behind the mirror (virtual images) are positive.
• Concave mirror: $u$ < 0, $f$ < 0, $r$ < 0. The sign of $v$ is negative for a real image and positive for a virtual image.
• Convex mirror: $u$ < 0, but $f$ > 0 and $r$ > 0; $v$ is always positive.
• Plane mirror: $f$ and $r$ are infinite; $u$ is negative and $v = -u$ (positive, behind the mirror).

This mirror equation calculator integrates the mirror formula, magnification relations, and sign convention into a single fast tool, making it easy to solve geometrical optics problems for any mirror.