Mirror Equation Calculator

About the Mirror Equation Calculator

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.

When This Page Helps

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.

How to Use the Inputs

1. Select concave or convex mirror type.
2. Enter the focal length OR the radius of curvature (R overrides f when provided).
3. Input the object distance in millimeters.
4. Optionally enter the object height for image height calculation.
5. Review image distance, magnification, and image classification.
6. Use the object distance table to explore different configurations.

Example Calculation

Tips & Best Practices

• Keep the sign convention consistent from start to finish because mirror problems are usually lost on sign errors, not arithmetic.
• If the object sits beyond the focal point of a concave mirror, expect a real inverted image; inside the focal point, expect a virtual upright image.
• Use radius of curvature only if you are sure the mirror is approximately spherical.
• Treat the result as paraxial optics first; off-axis rays and real mirror aberrations can shift the actual image quality.

Practical Guidance

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.

Common Pitfalls

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.

Frequently Asked Questions

• What is the sign convention for mirrors?

  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.

• What happens when the object is at the focal point?

  The reflected rays are parallel and the image forms at infinity. The mirror equation gives 1/d_i = 0, meaning d_i → ∞.

• Why do convex mirrors always produce virtual images?

  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.

• How is a shaving mirror different from a telescope 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.

• What is the difference between focal length and radius of curvature?

  R = 2f for a spherical mirror. The focal point is halfway between the mirror surface and the center of curvature.

• Does mirror equation work for parabolic mirrors?

  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.