Use the mirror equation (1/f = 1/do + 1/di) to find image properties for spherical mirrors. Determines whether the image is real or virtual, upright or inverted, and its magnification.
The mirror equation has the same mathematical form as the thin lens equation — 1/f = 1/d_o + 1/d_i — but applies to reflective surfaces instead of refractive ones. Concave (converging) mirrors and convex (diverging) mirrors are the two types you encounter regularly: shaving and makeup mirrors are concave, side-view "objects in mirror are closer than they appear" mirrors are convex, telescope primary mirrors are concave, and store anti-shoplifting mirrors are convex. Each follows the same equation but with characteristic sign conventions and image types.
This calculator handles both cases. Enter the focal length (or radius of curvature, where f = R/2) and the object distance, and it returns the image distance, magnification, and image properties (real or virtual, upright or inverted, enlarged or reduced). Use it for physics homework, telescope design (primary mirror specifications), or understanding how your car's side mirror compresses the wide view into the small reflective area.
The key sign convention: positive focal length means concave mirror (converging, can form real images); negative focal length means convex (diverging, only virtual images). Real images form in front of the mirror at positive d_i (where light actually converges); virtual images appear "behind" the mirror at negative d_i (where light only appears to come from).
**Scenario:** Concave shaving mirror with f = 10 cm. Your face is at d_o = 8 cm. What do you see? **Calculation:** 1/d_i = 1/10 − 1/8 = (4 − 5)/40 = −1/40. d_i = −40 cm. M = −(−40)/8 = +5. Image is virtual (behind mirror), upright, 5× enlarged. **Result:** Your face appears 5× larger and upright — exactly the effect of a magnifying shaving mirror. Useful for seeing fine detail (stubble, eyeliner) but you must stay closer than the focal length. Move too far back and the image flips and inverts (you cross d_o = f).
**Scenario:** Convex security mirror at a store ceiling with f = −20 cm (R = −40 cm). A person stands at d_o = 300 cm (3 m away). What does the camera see? **Calculation:** 1/d_i = 1/(−20) − 1/300 = −0.05 − 0.00333 = −0.0533. d_i = −18.75 cm. M = −(−18.75)/300 = +0.0625. Image: virtual, upright, ~1/16 actual size. **Result:** The mirror compresses a wide swath of the store into a small image, perfect for monitoring multiple aisles. Trade-off: distorted distance perception. Customers appear "smaller and farther" than they actually are, but the wide FOV is the point.
**Scenario:** A 1000mm focal length concave parabolic primary mirror. A distant star (d_o ≈ infinity). Where does the image form? **Calculation:** When d_o → ∞, 1/d_o → 0, so 1/d_i = 1/f → d_i = f = 1000 mm. Image forms at the focal point of the mirror. **Result:** Star's image forms exactly at f (1000 mm in front of the mirror). This is where you place the camera sensor or eyepiece. For terrestrial subjects (d_o much smaller), the image forms slightly past f (focus adjustment is needed). Maximum useful magnification of a telescope is limited by aperture (~50×/inch), not focal length.
**Use the mirror equation for:**
- **Telescope design**: primary and secondary mirror focal lengths, image plane location. - **Microscope mirror objectives**: high-mag reflective objectives for UV or IR. - **Shaving/makeup mirrors**: choosing focal length for appropriate magnification at typical viewing distance. - **Solar concentrators**: parabolic dish mirrors for steam generation, satellite communication antennas. - **Side-view mirrors and security mirrors**: convex for wide FOV, with understood distance distortion. - **Headlight reflectors**: parabolic mirrors with bulb at focal point produce parallel beam. - **Laser cavity design**: end mirrors of optical cavities.
**Image properties summary:**
**Concave (converging) mirror:**
- d_o > f: real, inverted image. Camera-like behavior. - d_o = f: parallel rays (image at infinity). - d_o < f: virtual, upright, enlarged image. Shaving-mirror behavior.
**Convex (diverging) mirror:**
- Always virtual, upright, reduced image. - Wider field of view than flat mirror. - "Objects in mirror are closer than they appear" — car side mirror trade-off.
**Common applications:**
- **Car side mirrors**: convex (~−1m focal length). Wide FOV, magnification ~0.5 at 5 m. - **Shaving/makeup mirrors**: concave (~10–15 cm focal length). 3–5× magnification at face distance. - **Telescope primaries**: large concave (often parabolic, not spherical). Focal lengths 0.5–10+ m. - **Dental mirrors**: small concave (~2 cm f), with object beyond f → real inverted image, magnified. - **Security mirrors at intersections**: convex (~30 cm f), give panoramic blind-spot view. - **Solar cookers**: parabolic concave, sun at infinity, image at focus = cooking pot.
**Vs. lens equation:**
The mirror equation and thin lens equation are mathematically identical (1/f = 1/d_o + 1/d_i), but the SIGN CONVENTION for d_i differs. For lenses, positive d_i = real image OPPOSITE side; for mirrors, positive d_i = real image SAME side as the object (both in front).
**Beyond simple spherical mirrors:**
- **Spherical aberration**: spherical mirrors don't focus parallel rays perfectly. Parabolic mirrors do, which is why telescope primaries are parabolic. - **Coma**: off-axis aberration in mirrors and lenses. Limits useful field of view in telescopes. - **Astigmatism**: also off-axis, more pronounced at fast f-numbers. - These aren't captured by the simple mirror equation; full ray-tracing is needed.
Calculate image distance, magnification, and image characteristics using the thin lens equation 1/f = 1/do + 1/di.
Calculate focal length from magnification, object/image distance, or field of view. Versatile focal length solver.
Optics-focused Snell's law with critical angle, Brewster's angle, total internal reflection, and Fresnel reflectance.
Calculate Brewster's angle where reflected light is fully polarized. Input refractive indices of two media.
Calculate optical absorbance, transmittance, and optical density using the Beer-Lambert law A = εbc.
Calculate the diffraction-limited angular resolution using the Rayleigh criterion θ = 1.22λ/D.
Positive for concave, negative for convex
Image Distance
24.00 cm
Magnification
-0.600×
Image
Real, Inverted
| Parameter | Value |
|---|---|
| Mirror Type | Concave |
| Focal Length f | 15.0000 cm |
| Radius of Curvature R | 30.0000 cm |
| Object Distance do | 40.0000 cm |
| Image Distance di | 24.0000 cm |
| Magnification M | -0.6000× |
| Image Type | Real |
| Orientation | Inverted |
| Formula | 1/f = 1/do + 1/di, f = R/2 |