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Put two mirrors at an angle, place a coin between them, and count the coins you see. The quick formula n = 360/θ − 1 gives the right answer for some angles and the wrong answer for others. This post shows why, gives the full rule, and explains the chain-of-reflections method that settles any case.
Is this in the syllabus?
NCERT Class 12 Chapter 9, Ray Optics does not treat inclined mirrors as a separate topic, though it mentions the kaleidoscope among optical devices. The JEE (Advanced) 2026 brochure lists "Reflection and refraction at plane and spherical surfaces", and counting images in two mirrors uses nothing more than repeated plane-mirror reflection. Check the current year's syllabus.
Why does the simple formula fail?
Lohani and Pathak (2023) surveyed school students and teachers and found that n = 360°/θ − 1, rounded to the nearest whole number, was widely taken to be the correct way to count images, whatever the angle or the object's position. The paper calls this a misconception and derives the correct count for any angle and position.
The reason is that images form in chains. Light from the object can hit mirror 1 first, then mirror 2, then mirror 1, and so on, or start with mirror 2. Each reflection makes a new image, but a chain stops when an image lands behind the next mirror it would need to reflect in. Where the chains stop depends on the angle and on where the object sits.
How do you count with chains?
All the images lie on a circle centred on the line where the mirrors meet, at the object's distance from that line. Measure angles from mirror 1, and put mirror 2 at angle θ and the object at angle α between them.
Reflection in mirror 1 sends an angle φ to −φ.
Reflection in mirror 2 sends φ to 2θ − φ.
An image can be reflected again only if it lies in front of that mirror.
Worked example 1: θ = 90°
m = 360/90 = 4, even, so expect 3 images. With α = 30°: chain starting at mirror 1 gives −30°, then 2(90) + 30 = 210° (the same as −150°). Chain starting at mirror 2 gives 150°, then −150°, which is already counted. Images at −30°, 150° and 210°: three, as expected. The image at 210° is the one you see in the corner, formed by both mirrors.
Worked example 2: θ = 72°, two positions
m = 360/72 = 5, odd.
Object on the bisector, α = 36°: the chains give images at −36°, 108°, −108° and 180°. Two chains meet at 180°, so there are 4 images (m − 1).
Object off the bisector, α = 20°: the chains give −20°, 164° and −164° from one side, and 124° and −124° from the other. That is 5 images (m), because the two chains no longer end on the same point.
Worked example 3: θ = 50°
m = 360/50 = 7.2, not a whole number. The rounded formula would say 6 (or 7 if you round m first). Counting with chains:
Object on the bisector, α = 25°: 8 images.
Object at α = 10°: 7 images.
So the same pair of mirrors gives a different number of images as the object moves. This is the case the 2023 paper says students and teachers most often get wrong.
What about parallel mirrors?
When θ = 0 the chains never end, so in principle there are infinitely many images. In practice each reflection loses some light, so the images fade with distance.
Which mistakes cost marks?
Using 360/θ − 1 for every angle. It is right only when 360/θ is even, or odd with the object on the bisector.
Rounding a non-integer m. Count instead.
Ignoring the object's position. For odd and non-integer m it matters.
Counting a coincident image twice. When two chains reach the same point, it is one image.
A practice routine
Draw a circle and two mirror lines at 60°. Place an object at 20° and count by chains; check you get 5.
Repeat at 120° with the object on and off the bisector (2 and 3).
Try 45°, then a non-integer case such as 50°, at two positions.
Review plane-mirror basics in our post on common ray optics mistakes.
If the chain method feels hard to picture, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
Next step: stand two hand mirrors at about 72°, move a coin along the gap, and count the images at the bisector and near one mirror.
