Mirror Equation Calculator
Solve the mirror equation for focal length, object distance, image distance, magnification, image height, and real or virtual image type.
Physics calculator for spherical mirrors
This mirror equation calculator uses 1/f = 1/do + 1/di to solve for focal length, object distance, or image distance. It also calculates magnification, image height, image orientation, and whether the image is real or virtual.
Use a positive focal length for a concave mirror and a negative focal length for a convex mirror. Object distance is usually positive when the object is in front of the mirror.
The calculator follows the common classroom sign convention for spherical mirrors; always match the sign convention used by your teacher, textbook, or lab.
Image distance
Image is -- and --
Focal length
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--
Object distance
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Image distance
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--
Magnification
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--
Image height
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Mirror type
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Ray diagram clue
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Simple mirror visual
Object
--
Image
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Note: If the object is at the focal point of a concave mirror, the reflected rays are parallel and the image distance is effectively infinite. The calculator will warn when the denominator is near zero.
How to use the mirror equation calculator
- Choose the unknown: Solve for image distance, focal length, or object distance.
- Select mirror type: Choose concave, convex, or signed focal length if your problem already gives a signed value.
- Enter known distances: Use the same unit for object distance, image distance, focal length, and object height.
- Check signs: Real images usually have positive image distance, while virtual images usually have negative image distance.
- Review magnification: A negative magnification means inverted, while a positive magnification means upright.
Mirror equation formula
The mirror equation connects focal length, object distance, and image distance for spherical mirrors. It is usually written as 1/f = 1/do + 1/di.
A mirror equation calculator finds image distance, object distance, or focal length using 1/f = 1/do + 1/di. Enter two known values to solve the third. Concave mirrors use positive focal length, convex mirrors use negative focal length, and distances follow the chosen sign convention.
Magnification compares image height with object height. For mirrors, m = -di / do, and image height equals magnification times object height.
1/f = 1/do + 1/di
m = -di / do
hi = m x ho
Example: if a concave mirror has f = 10 cm and an object is 30 cm in front of the mirror, the image distance is 15 cm and the magnification is -0.5.
Formula reference: OpenStax University Physics - Spherical Mirrors.
Mirror sign convention table
| Quantity | Positive sign | Negative sign | What it usually means |
|---|---|---|---|
| Focal length f | Concave mirror | Convex mirror | Concave mirrors can form real images; convex mirrors form virtual images. |
| Object distance do | Object in front of mirror | Virtual object in special setups | Most classroom object distances are positive. |
| Image distance di | Real image | Virtual image | Positive image distance is in front of the mirror. |
| Magnification m | Upright image | Inverted image | Magnitude greater than 1 means enlarged. |
Sign convention reference: The Physics Classroom - The Mirror Equation.
Concave mirror object-position guide
| Object position | Image position | Image type | Magnification clue | Calculator check |
|---|---|---|---|---|
| Beyond center of curvature (do > 2f) | Between focal point and center of curvature | Real and inverted | Reduced, |m| < 1 | di is positive and smaller than do. |
| At center of curvature (do = 2f) | At center of curvature | Real and inverted | Same size, m = -1 | di equals do with opposite height sign. |
| Between center and focal point (f < do < 2f) | Beyond center of curvature | Real and inverted | Enlarged, |m| > 1 | di is positive and larger than do. |
| At focal point (do = f) | No finite image distance | Rays leave parallel | Undefined | The equation denominator is zero. |
| Inside focal point (do < f) | Behind the mirror | Virtual and upright | Enlarged, m is positive | di is negative. |
Ray diagram construction guide
The mirror equation gives the number, while a ray diagram shows whether the number makes visual sense. Use these principal rays as a quick diagram checklist.
Parallel ray
Draw a ray parallel to the principal axis. For a concave mirror it reflects through the focal point; for a convex mirror it reflects as if it came from the focal point.
Focal ray
Draw a ray through the focal point before reflection. After reflection, it travels parallel to the optical axis.
Center ray
Draw a ray through the center of curvature. It strikes the mirror along a radius and reflects back along the same path.
Real images form where reflected rays actually meet. Virtual images form where reflected rays only appear to meet when traced backward behind the mirror.
Mirror equation answer-check workflow
Before submitting a homework answer or lab result, use these checks to catch common sign, unit, and interpretation errors.
1. Confirm the mirror type
Concave mirrors use positive focal length in this convention. Convex mirrors use negative focal length and should give a virtual image for a real object.
2. Check radius of curvature
If the problem gives radius of curvature, convert with f = R / 2 before entering focal length. Keep the same unit throughout.
3. Compare signs to image type
Positive image distance should match a real image; negative image distance should match a virtual image.
4. Compare magnification to diagram
Negative magnification should look inverted in the ray diagram, while positive magnification should look upright.
Real, virtual, upright, and inverted images
The signs in the mirror equation describe where the image forms and how it looks. These labels are often the fastest way to check whether an answer makes physical sense.
Real image
di is positive, and the image can be projected on a screen.
Virtual image
di is negative, and the image appears behind the mirror.
Orientation
Positive magnification is upright; negative magnification is inverted.
Common mirror equation mistakes
Most wrong answers come from signs, mixed units, or using the lens equation convention without checking whether the mirror convention is the same.
Forgetting convex f is negative
A convex mirror has negative focal length in the common mirror sign convention.
Mixing units
Do not enter focal length in cm and object distance in meters unless you convert first.
Ignoring the focal point
When an object is at the focal point of a concave mirror, no finite image distance is produced.
Interesting fact
NASA's James Webb Space Telescope uses 18 hexagonal mirror segments that are aligned to act as one large 6.5-meter mirror. That segmented design is a real-world example of how precise mirror shape and alignment control image formation in geometric optics. Source: NASA Science - Webb's Mirrors.
Frequently Asked Questions
What is the mirror equation in geometric optics?
The mirror equation is 1/f = 1/do + 1/di. It connects focal length, object distance, and image distance for spherical mirrors in geometric optics. If you know two of the three values, the calculator can solve the third and help check image formation in a physics problem.
Is focal length positive or negative for concave and convex mirrors?
In the common mirror sign convention, a concave mirror has a positive focal length and a convex mirror has a negative focal length. The focal point lies on the principal axis, also called the optical axis, and is related to the center of curvature and radius of curvature by f = R/2. Some courses use different conventions, so check your class notes or textbook before finalizing an answer.
What does a negative image distance mean?
A negative image distance usually means the image is virtual and appears behind the mirror. A convex mirror always forms a virtual image for a real object, while a concave mirror can form a virtual image when the object distance is shorter than the focal length. A positive image distance usually indicates a real image in front of the mirror.
How do I calculate magnification and image height?
Use the formula m = -di / do, then multiply magnification by object height to get image height. If magnification is negative, the result is an inverted image. If it is positive, the result is an upright image. If the magnitude is greater than 1, the image is enlarged; if it is less than 1, the image is reduced.
What happens when the object is at the focal point?
For a concave mirror, an object at the focal point sends reflected rays out parallel to each other along the principal axis. The mirror equation denominator becomes zero, so there is no finite image distance. In a ray diagram, the image is often described as being at infinity.
Can I use the mirror equation like a lens equation?
The equation shape is similar to the thin lens equation, but the sign convention and physical meaning can differ. Mirrors reflect light, while lenses refract it, so focal length, image distance, and real or virtual image labels may be interpreted differently. Use a lens equation calculator or your textbook's lens convention when working with lenses instead of mirrors, and keep unit conversion consistent.
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Mirror equation calculator disclaimer
This mirror equation calculator is for general educational and homework-planning purposes only. It is not a substitute for physics instruction, lab measurement, optical engineering, safety advice, medical optics advice, professional design work, or textbook-specific grading requirements.
Sign conventions can vary by course, textbook, teacher, or optical system. Results depend on the values and signs you enter, the spherical mirror approximation, paraxial rays, and consistent distance units. Real mirrors may show aberration, alignment errors, manufacturing tolerances, or measurement uncertainty.
Do not rely on this calculator for safety-critical optical systems, eye-care decisions, laser work, vehicle mirrors, surveillance systems, scientific instruments, or engineering specifications. Confirm important work with your instructor, textbook, lab procedure, qualified physics professional, or optical engineer.
Last updated: May 17, 2026.