Mirror Equation Calculator
Free calculate focal length, object distance, or image distance using 1/f = 1/d₀ + 1/dᵢ. Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Positive for real object (in front of mirror)
Positive for real image, negative for virtual
Results
Enter parameters to calculate
What is the Mirror Equation?
The mirror equation relates the focal length (f), object distance (d₀), and image distance (dᵢ) of a mirror: 1/f = 1/d₀ + 1/dᵢ.
This equation applies to both concave (converging) and convex (diverging) mirrors. The sign conventions are: focal length is positive for concave mirrors and negative for convex mirrors; object distance is positive for real objects; image distance is positive for real images (in front of mirror) and negative for virtual images (behind mirror).
The mirror equation is fundamental in geometric optics, used to predict where images form, their size, and whether they're real or virtual. It's essential for understanding how mirrors work in telescopes, microscopes, and everyday applications.
Mirror Equation Formula
Where:
- • f = Focal length (cm or m) - positive for concave, negative for convex
- • d₀ = Object distance (cm or m) - positive for real object
- • dᵢ = Image distance (cm or m) - positive for real image, negative for virtual
Magnification: m = -dᵢ/d₀ = hᵢ/h₀
Positive m = upright image, negative m = inverted image
How to Calculate
-
1
Identify what to solve for
Determine if you need focal length, object distance, or image distance.
-
2
Apply sign conventions
Use positive f for concave, negative for convex. Positive d₀ for real objects. Positive dᵢ for real images, negative for virtual.
-
3
Rearrange the equation
If solving for f: 1/f = 1/d₀ + 1/dᵢ, then f = 1/(1/d₀ + 1/dᵢ). For dᵢ: 1/dᵢ = 1/f - 1/d₀. For d₀: 1/d₀ = 1/f - 1/dᵢ.
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4
Calculate and interpret
Solve for the unknown. Check the sign to determine if image is real/virtual, upright/inverted.
Practical Examples
Example 1: Concave Mirror
f = 10 cm, d₀ = 20 cm. Find image distance.
Solution:
1/dᵢ = 1/f - 1/d₀ = 1/10 - 1/20 = 1/20
dᵢ = 20 cm (real image, inverted)
Example 2: Convex Mirror
f = -15 cm, d₀ = 30 cm. Find image distance.
Solution:
1/dᵢ = 1/(-15) - 1/30 = -1/10
dᵢ = -10 cm (virtual image, upright)
Applications
Telescopes
Designing reflecting telescopes using concave mirrors to focus light from distant objects.
Automotive
Side-view mirrors use convex mirrors to provide wider field of view (virtual, reduced images).
Optics
Understanding image formation in mirrors, designing optical systems, and analyzing light reflection.
Education
Teaching geometric optics, understanding reflection, and demonstrating image formation principles.
Frequently Asked Questions
What's the difference between concave and convex mirrors?
Concave mirrors (f > 0) converge light and can form real or virtual images. Convex mirrors (f < 0) always diverge light and form only virtual, upright, reduced images.
When is the image real vs. virtual?
Real images (dᵢ > 0) form in front of the mirror where light actually converges. Virtual images (dᵢ < 0) form behind the mirror where light appears to come from but doesn't actually converge.
How do I calculate magnification?
Magnification m = -dᵢ/d₀ = hᵢ/h₀. Negative m means inverted image, positive m means upright. |m| > 1 means enlarged, |m| < 1 means reduced.
What if the object is at the focal point?
If d₀ = f, then 1/dᵢ = 0, so dᵢ = ∞. The image forms at infinity - rays are parallel after reflection. This is used in searchlights and car headlights.
Can the mirror equation be used for plane mirrors?
For plane mirrors, f = ∞, so 1/f = 0. The equation gives dᵢ = -d₀ (virtual image same distance behind mirror as object is in front). Magnification m = 1 (same size, upright).