Mirror Equation Calculator — Focal Length, Object & Image Distance
Solve the spherical mirror equation for any unknown — focal length, object distance or image distance — and get the magnification and image type in one step.
cm
cm
cm
Positive → real image in front; negative → virtual image behind mirror
Mirror equation
Solve for dᵢ
Result
Magnification
- 1
1/f — inverse focal length
1 ÷ 15 = 0.066667 - 2
1/dₒ — inverse object distance
1 ÷ 30 = 0.033333 - 3
1/dᵢ = 1/f − 1/dₒ
0.066667 − 0.033333 = 0.033333Rearranging the mirror equation 1/f = 1/dₒ + 1/dᵢ. - 4
Image distance dᵢ
1 ÷ (0.033333) = 30
How does this calculator work?
The mirror equation 1/f = 1/dₒ + 1/dᵢ relates focal length, object distance and image distance for spherical mirrors. Magnification is m = −dᵢ/dₒ (negative = inverted). Positive dᵢ = real image in front of the mirror; negative dᵢ = virtual image behind it. Enter any two quantities to solve for the third.
Formula
How this is calculated
For a spherical mirror the relationship between focal length f, object distance dₒ and image distance dᵢ is given by the mirror equation 1/f = 1/dₒ + 1/dᵢ. Knowing any two of these three quantities lets you solve for the third by simple algebraic rearrangement: dᵢ = 1/(1/f − 1/dₒ), dₒ = 1/(1/f − 1/dᵢ), or f = 1/(1/dₒ + 1/dᵢ).
The sign convention used here (real-is-positive) assigns positive values to distances measured in front of the mirror (the side the light comes from) and negative values to distances measured behind it. Concave (converging) mirrors have positive focal lengths; convex (diverging) mirrors have negative focal lengths. A positive image distance means a real, inverted image forms in front of the mirror; a negative image distance means a virtual, upright image appears behind it.
Magnification m = −dᵢ/dₒ gives the ratio of image height to object height: |m| > 1 means the image is larger than the object, |m| < 1 means it is smaller, and the sign (negative → inverted, positive → upright) tells you orientation. The radius of curvature R = 2f connects the formula to the mirror geometry — a mirror with a radius of 30 cm has a focal length of 15 cm.
Frequently asked questions
A negative dᵢ means the image is virtual — it appears to be behind the mirror and cannot be projected onto a screen. Virtual images are always upright and are seen by looking into the mirror, as with a convex car mirror.
A concave mirror produces a real, inverted image when the object is placed beyond the focal point (dₒ > f). When the object is between the focal point and the mirror (dₒ < f), the image is virtual, upright and magnified — this is how a magnifying shaving/make-up mirror works.
The thin-lens equation has the same mathematical form (1/f = 1/dₒ + 1/dᵢ) but uses a different sign convention — for lenses, positive dᵢ is on the opposite side from the object (transmission side). The physics and algebra are analogous but the sign rules differ.
Also known as
TG we-Calculate Editorial Team. (2026). Mirror Equation Calculator — Focal Length, Object & Image Distance [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/mirror-equation-calculator
TG we-Calculate Editorial Team. "Mirror Equation Calculator — Focal Length, Object & Image Distance." TG we-Calculate. 2026. https://we-calculate.com/calculator/mirror-equation-calculator.
TG we-Calculate Editorial Team, "Mirror Equation Calculator — Focal Length, Object & Image Distance," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/mirror-equation-calculator
@misc{wecalculate_mirror_equation_calculator, title = {Mirror Equation Calculator — Focal Length, Object & Image Distance}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/mirror-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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