Calculate image distance, magnification, and image properties for concave and convex mirrors using the mirror equation 1/f = 1/d_o + 1/d_i.
The mirror equation 1/f = 1/d_o + 1/d_i is the fundamental relationship governing image formation by curved mirrors. For concave (converging) mirrors, the focal length is positive; for convex (diverging) mirrors, it is negative. Depending on the object's position relative to the focal point, the image can be real or virtual, upright or inverted, enlarged or diminished.
Curved mirrors are essential components in telescopes (Newtonian and Cassegrain designs), car side mirrors ("objects may be closer"), solar concentrators, satellite dishes, shaving/makeup mirrors, and laser cavities. The radius of curvature R equals twice the focal length (R = 2f), and the magnification m = −d_i/d_o determines both the size and orientation of the image.
This calculator handles both concave and convex mirrors, accepts either focal length or radius of curvature, and computes image distance, magnification, image height, image type (real/virtual, upright/inverted, enlarged/diminished), and optical power. A comprehensive table shows image behavior for objects at various multiples of the focal length, and a simplified ray diagram provides visual context for the mirror-object-image geometry.
Use this calculator when you need to classify the image from a curved mirror without rebuilding the sign convention each time.
It is useful for basic optics work, telescope examples, shaving and makeup mirrors, and any setup where object position relative to the focal point changes the image behavior. It keeps the focal length, image distance, and magnification together so the image type can be checked in one pass.
1/f = 1/d_o + 1/d_i. Magnification: m = −d_i/d_o. Image height: h_i = m × h_o. Radius: R = 2f. For convex mirrors: f is negative.
Result: Image at 100 mm, m = −1.0 (real, inverted, same size)
Object at 2f for a concave mirror: 1/50 = 1/100 + 1/d_i → d_i = 100 mm. Image is real, inverted, and same size — this is the special case of an object at the center of curvature.
Mirror problems become much easier when you think in regions: object beyond 2f, at 2f, between f and 2f, at f, and inside f. Each region has a characteristic image type, so the equation becomes a way to quantify the result rather than to discover it from scratch.
Most mistakes come from mixing up the sign of focal length and image distance. Another common issue is treating every curved mirror like a perfect paraxial mirror; spherical aberration and off-axis geometry can matter in real optical systems even when the thin mirror equation itself is correct.
For real objects and real images, distances are positive (in front of the mirror). Virtual images have negative image distance (behind the mirror). Concave f > 0, convex f < 0.
The reflected rays are parallel and the image forms at infinity. The mirror equation gives 1/d_i = 0, meaning d_i → ∞.
With f < 0 and d_o > 0, the equation 1/d_i = 1/f − 1/d_o is always negative, so d_i < 0 (virtual). The image is always upright, diminished, and behind the mirror.
Both are concave. A shaving mirror places your face inside the focal length to produce an enlarged virtual image. A telescope mirror uses objects far beyond 2f to create real, diminished images.
R = 2f for a spherical mirror. The focal point is halfway between the mirror surface and the center of curvature.
Yes, for on-axis rays. Parabolic mirrors eliminate spherical aberration, so the equation is exact on-axis. For off-axis points, more complex analysis is needed.