Mirror Equation Calculator
Enter a focal length and one distance, or leave the focal length blank and enter both distances, to solve the spherical mirror equation 1/f = 1/do + 1/di. The calculator returns the focal length, the radius of curvature (R = 2f), the missing distance, the magnification m = -di/do, and whether the image is real and inverted or virtual and upright. It follows the real-is-positive sign convention, so concave mirrors take positive focal lengths and convex mirrors take negative ones.
How to use the mirror equation calculator
- Enter the focal length f, using a positive value for a concave mirror or a negative value for a convex mirror.
- Type the object distance do (an object in front of the mirror is positive).
- To solve for focal length instead, clear the focal length field and enter both do and the image distance di.
- Read the solved focal length, radius R, image distance and magnification in the result cards.
- Check the real-inverted or virtual-upright label, then click the copy button to save the full result.
Examples
Concave mirror, real image
f = 10, do = 30
di = 15, R = 20, m = -0.5 (real, inverted)
Solve focal length from two distances
do = 30, di = 15 (focal length blank)
f = 10, R = 20, m = -0.5 (real, inverted)
Convex mirror, virtual image
f = -10, do = 30
di = -7.5, R = -20, m = 0.25 (virtual, upright)
Frequently asked questions
What is the mirror equation?
For a spherical mirror, 1/f = 1/do + 1/di, where f is the focal length, do the object distance and di the image distance. Given any two of those three quantities, the calculator solves for the third and reports the magnification alongside it.
How are focal length and radius of curvature related?
The focal length is half the radius of curvature, f = R/2, so R = 2f. Enter the focal length and the tool computes the radius, or work back from a radius you already know by dividing it by two.
What sign convention does this use?
Real is positive. A concave (converging) mirror has a positive focal length and radius, while a convex (diverging) mirror has negative values. A real image formed in front of the mirror has a positive image distance; a virtual image behind the mirror is negative.
How is magnification calculated?
Magnification m = -di / do. A negative value means the image is inverted and a positive value means it is upright, while the absolute value is the height ratio. So m = -0.5 means the image is half the object's height and flipped over.
What do the real-inverted and virtual-upright labels mean?
The calculator labels the result from the sign of the image distance: a positive di is a real, inverted image that could be projected onto a screen, and a negative di is a virtual, upright image seen only by looking into the mirror. Concave mirrors can produce either case depending on object position, while convex mirrors always give virtual, upright images.
Can I enter a radius of curvature directly?
The on-page form has focal length, object distance and image distance fields, so enter half the radius as the focal length. The underlying engine and JSON API also accept a radiusOfCurvature field if you call the API instead.
Why do I get an error when the object sits at the focal point?
If the object distance equals the focal length, 1/f - 1/do is zero and the rays leave parallel, so the image forms at infinity and no finite distance exists. Move the object off the focal point to get a numeric answer.
Does the tool care what units I use?
No. The equation is unit-agnostic, so as long as every distance uses the same unit (all centimetres, or all metres) the focal length, radius and image distance come back in that same unit. Magnification is a pure ratio with no units.
Is my input sent anywhere?
No. The calculation runs locally in your browser, so nothing you type is uploaded or stored. It works offline once the page has loaded.
Learn more
- Concave vs convex mirror images
How image type, size and orientation change with object position for concave and convex mirrors, and how the signs map to the math.
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