Mirror Equation Calculator

The mirror equation relates the focal length, object distance, and image distance for spherical mirrors:

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$

Rearranging to solve for image distance:

$$d_i = \frac{f \cdot d_o}{d_o - f}$$

The sign conventions for mirrors differ from lenses:

• Concave mirror: f is positive (converging)
• Convex mirror: f is negative (diverging)
• d_o is always positive for real objects
• d_i > 0: real image (in front of mirror)
• d_i < 0: virtual image (behind the mirror)

The magnification is:

$$M = -\frac{d_i}{d_o}$$

For a concave mirror, the image properties depend on object position relative to the focal point and center of curvature. When the object is beyond the center of curvature (d_o > 2f), the image is real, inverted, and diminished. Between the center and focal point (f < d_o < 2f), the image is real, inverted, and magnified. Inside the focal point (d_o < f), the image is virtual, upright, and magnified — the principle behind makeup mirrors.

For a convex mirror, the image is always virtual, upright, and diminished regardless of object position. This wide-angle property makes convex mirrors ideal for security mirrors and vehicle side mirrors, which carry the warning "objects in mirror are closer than they appear."

The relationship between focal length and radius of curvature is $$f = \frac{R}{2}$$ for spherical mirrors under the paraxial approximation, where rays are close to and nearly parallel to the optical axis.

The signed focal length shows the effective focal length with the correct sign convention — positive for concave and negative for convex mirrors. The image distance tells you where the image forms: positive values mean a real image in front of the mirror, negative values mean a virtual image behind the mirror surface.

The magnification indicates both size and orientation. A magnification of −0.5 means the image is half-size and inverted. A magnification of +3 means the image is three times larger and upright (virtual). The image type description summarizes all these properties in plain language.