Mirror Equation Calculator
Enter object distance and radius of curvature (with optional object height) to compute image distance and magnification.
Input Data
Results
At a glance:The mirror equation for a spherical mirror is 1/f = 1/d_o + 1/d_i, with focal length f = R/2 (R radius of curvature). Sign conventions: for concave mirrors R>0 (real focus), for convex mirrors R<0 (virtual focus). The magnification m = h_i/h_o = −d_i/d_o; a negative m means the image is inverted. Real images form on the same side as the reflecting surface (d_i>0); virtual images form behind the mirror (d_i<0). This calculator handles object distance, radius, and object height to give image distance and height.
Formula
Mirror equation: 1/f = 1/d_o + 1/d_i
Focal length: f = R/2
Magnification: m = −d_i/d_o = h_i/h_o
$$f = \frac{R}{2}$$$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$$$v = \frac{uf}{u - f}$$$$m = -\frac{v}{u}$$How to Use
- Enter the object distance d_o (cm).
- Enter the radius of curvature R (cm; negative for convex).
- Optionally enter object height h_o (cm).
- The calculator returns image distance d_i, image height and magnification.
Case Studies
Concave makeup mirror
R = 40 cm (f = 20 cm), d_o = 10 cm.
1/d_i = 1/20 − 1/10 = −1/20 → d_i = −20 cm (virtual, behind mirror).
m = −(−20)/10 = 2: upright, magnified 2×.
FAQ
Why is f = R/2?
For a spherical mirror of radius R, rays parallel to the axis reflect through the focal point halfway between the mirror vertex and its center of curvature, so f = R/2.
What does a negative image distance mean?
It means the image is virtual and forms behind the mirror (typical for convex mirrors and for objects inside the focal length of a concave mirror). The image is upright (m>0).
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.