About the Mirror Equation Calculator
This mirror equation calculator solves 1/f = 1/dₒ + 1/dᵢ for a spherical mirror. Choose a concave, convex or plane mirror, give its focal length or radius of curvature, and enter how far the object is from the mirror. It returns where the image forms, the magnification, the image height and whether the image is real or virtual, upright or inverted, larger or smaller.
Students use it to check ray diagrams and optics homework; it also explains everyday mirrors — why a shaving or makeup mirror (concave) magnifies your face up close, why a car’s passenger-side mirror (convex) makes objects look smaller and farther away, and how a reflecting telescope forms a real image. The table lists where the image lands for a range of object positions.
The calculator uses the real-is-positive sign convention: object and image distances in front of the mirror are positive, an image behind the mirror is negative (virtual), concave mirrors have a positive focal length and convex mirrors a negative one. Enter the focal length or radius as a positive size; the sign is applied for you from the mirror type. The model assumes paraxial rays (small angles) and ignores spherical aberration.
With the default inputs, the image distance (dᵢ) is 15 cm. Change any value above to recalculate instantly.
How to use the mirror equation calculator
- 1Choose concave, convex or plane mirror.
- 2Say whether you know the focal length or the radius, and enter its size.
- 3Enter the object distance from the mirror.
- 4Optionally enter the object height to get the image height.
- 5Read the image distance, magnification and image type.
Formula and method
The mirror equation relates the focal length of a spherical mirror to the object and image distances measured from the mirror’s surface. Rearranged for the image, dᵢ = 1 ÷ (1/f − 1/dₒ). The focal length is half the radius of curvature. Using the real-is-positive convention, concave mirrors have f > 0 and convex mirrors f < 0; a positive dᵢ means a real image in front of the mirror and a negative dᵢ a virtual image behind it.
Magnification is m = −dᵢ ÷ dₒ, and the image height is m × hₒ. A negative m means the image is inverted; |m| > 1 means it is enlarged. A plane mirror has an infinite focal length, so dᵢ = −dₒ and m = +1. The equation is exact only for rays close to the mirror’s axis; wide-aperture spherical mirrors blur slightly (spherical aberration).
- f
- Focal length (+ concave, − convex)
- R
- Radius of curvature
- dₒ
- Object distance from the mirror
- dᵢ
- Image distance (+ real, in front; − virtual, behind)
- m
- Magnification (− inverted, + upright)
Worked examples
Object 30 cm from a concave mirror with f = 10 cm
1/dᵢ = 1/10 − 1/30 = 2/30, so dᵢ = 15 cm in front of the mirror. m = −15 ÷ 30 = −0.5, so a 5 cm object gives a real, inverted image 2.5 cm tall.
Makeup mirror: face 5 cm from a 10 cm concave mirror
1/dᵢ = 1/10 − 1/5 = −1/10, so dᵢ = −10 cm: a virtual image behind the mirror, upright and magnified 2×.
Convex security mirror, f = 10 cm, object at 30 cm
For a convex mirror f = −10 cm, so 1/dᵢ = −1/10 − 1/30 = −4/30 and dᵢ = −7.5 cm. The image is virtual, upright and one quarter size.
Concave mirror with radius 40 cm, object at 60 cm
f = R ÷ 2 = 20 cm. 1/dᵢ = 1/20 − 1/60 = 2/60, so dᵢ = 30 cm and m = −0.5: a real, inverted image half the size.
Frequently asked questions
What is the mirror equation?+
The mirror equation is 1/f = 1/dₒ + 1/dᵢ, where f is the focal length, dₒ the object distance and dᵢ the image distance. It is the same form as the thin lens equation.
Is the focal length of a convex mirror negative?+
Yes, in the real-is-positive convention. A convex mirror diverges light and its focal point is behind the mirror, so f is negative and the image distance is always negative (virtual).
When does a concave mirror form a virtual image?+
Only when the object is closer to the mirror than the focal point. The image is then virtual, upright and magnified. Beyond the focal point the image is real and inverted.
How is the focal length related to the radius of curvature?+
For a spherical mirror the focal length is half the radius of curvature, f = R/2. A mirror ground from a sphere of 40 cm radius focuses parallel light 20 cm in front of it.
Why do car mirrors say objects are closer than they appear?+
Passenger-side mirrors are often convex. They form reduced virtual images close behind the mirror, which widens the view but makes vehicles look smaller and therefore farther away than they really are.