Hong Life CalculatorsUnderstanding the mirror equation calculator — concept and principle
This calculator applies the mirror equation to concave, convex and plane mirrors. From the focal length, object distance and object height it gives the image distance, magnification, image height and whether the image is real or virtual, upright or inverted, larger or smaller. It is used in optics courses and for designing reflectors, telescopes and shaving mirrors.
How it is calculated
1/d_o + 1/d_i = 1/f, with f = R/2
m = h_i ÷ h_o = −d_i ÷ d_o
- f and R are positive for a concave mirror and negative for a convex one.
- d_i positive means a real image in front of the mirror; negative means a virtual image behind it.
- Negative m means the image is inverted. A plane mirror gives d_i = −d_o and m = 1.
The principle
A concave mirror reflects rays parallel to its axis through a focal point halfway between the mirror and its centre of curvature. Rays from each point of an object therefore meet again at a definite image point, and simple geometry with similar triangles gives the mirror equation. A convex mirror spreads rays out so they only appear to come from behind it, giving a virtual, upright, smaller image. The equation holds for rays close to the axis.
Worked example
Default inputs: concave mirror, |f| = 15 cm, object at 40 cm, object height 5 cm.
- 1/d_i = 1/15 − 1/40 = 5/120, so d_i = 24 cm.
- m = −24 ÷ 40 = −0.6.
- Image height = −0.6 × 5 = −3 cm.
- d_i is positive, so the image is real, in front of the mirror, and m is negative and less than 1: inverted, reduced.
- R = 2 × 15 = 30 cm; signed f = 15 cm.
| Focal length |f| (cm) | Image distance from the mirror (cm) |
|---|---|
| 3.75 | 4.14 |
| 7.5 | 9.23 |
| 11.2 | 15.65 |
| 15 | 24 |
| 18.8 | 35.29 |
| 22.5 | 51.43 |
| 30 | 120 |
How to read the result
With a concave mirror, an object beyond 2f gives a smaller inverted real image, between f and 2f a larger inverted one, and inside f a larger upright virtual image, which is how shaving and make-up mirrors magnify. A convex mirror always gives a small upright virtual image with a wide field of view, which is why it is used for car side mirrors and shop security.
Common mistakes
- Using the wrong sign for a convex mirror's focal length.
- Putting the object exactly at the focal point, where no image forms.
- Reading a negative image height as an error; it means the image is inverted.
Frequently asked questions
How does a make-up mirror magnify?
It is concave, and your face sits inside its focal length, giving an enlarged upright virtual image.
Why are objects in car side mirrors closer than they appear?
The convex mirror makes small images, which look farther away.
What is a real image?
One where light actually converges, so it can be projected on a screen.
What happens at the focal point?
Reflected rays are parallel and no image forms.
Sources: OpenStax, University Physics Volume 3 (2016), section 2.2 "Spherical Mirrors"