Mirror Equation Calculator

Solve the mirror equation for concave and convex mirrors. Find image position, magnification, and focal length using the New Cartesian sign convention.

🪞 Mirror Equation Calculator
Mirror Type
Object Distance (u)30.0 cm
cm
1 cm200 cm
Focal Length (f)20.0 cm
cm
1 cm100 cm
Object Height (optional)
cm
Mirror Type
Object Distance (u)30.0 cm
cm
1 cm200 cm
Image Distance (v)60.0 cm
cm
1 cm200 cm
Image Type
Image Distance
—
Formula Used
—
Image Distance (|v|)
—
Image Type
—
Orientation / Size
—
Magnification (m)
—
Image Height
—
Focal Length (f)
—
Radius of Curvature (R)
—

🪞 What is the Mirror Equation?

The mirror equation is the fundamental formula for spherical mirrors (both concave and convex) that relates the object distance (u), image distance (v), and focal length (f): 1/v + 1/u = 1/f. It allows you to find where an image forms, whether it is real or virtual, and how it is magnified, given any two of the three quantities. The companion formula for magnification is m = minus v divided by u, which tells you both the size ratio and orientation (upright or inverted) of the image relative to the object.

The mirror equation has many real-world applications. Concave mirrors are used in car headlights and searchlights where the light source is placed at the focal point, producing a parallel reflected beam. The same principle works in reverse for satellite dish antennas. Shaving and makeup mirrors are concave mirrors with the face placed inside the focal length, producing a magnified virtual image. Convex mirrors are used as rear-view mirrors in vehicles and as security mirrors in stores because they give a wide field of view, always producing a virtual, upright, and diminished image regardless of object distance. Dental mirrors are small concave mirrors that dentists use to see magnified reflections of tooth surfaces.

A common confusion is the sign convention. This calculator uses the New Cartesian Sign Convention, which is the standard taught in NCERT Class 10 and Class 12 physics and adopted across most Indian and UK school curricula. In this convention, all distances are measured from the pole (vertex) of the mirror. Distances measured in the direction of incident light are positive; distances in the opposite direction are negative. For a real object placed in front of a mirror, u is always negative. For a concave mirror, f is negative (focal point is in front of the mirror, same side as the object). For a convex mirror, f is positive (focal point is behind the mirror, virtual). To keep data entry simple, this calculator accepts the absolute values of u and f as positive numbers and applies the correct signs internally based on the mirror type you select.

This calculator solves both common problems: finding image position from a known object position and focal length (Find Image mode), and finding the focal length of an unknown mirror from known object and image positions (Find Focal Length mode). It also accepts optional object height to compute image height, and displays whether the image is real or virtual, upright or inverted, and magnified or diminished, which are the four characteristics students need to fully describe a mirror image.

📐 Formula

1/v  +  1/u  =  1/f    and    m  =  −v/u  =  hi/ho
u = object distance from pole (negative for real objects; users enter absolute value)
v = image distance from pole (negative = real, in front; positive = virtual, behind)
f = focal length (negative for concave; positive for convex)
R = radius of curvature = 2f
m = magnification (negative = inverted image; positive = upright image)
hi = image height; ho = object height; hi = m × ho
Sign Convention (New Cartesian): Object in front → u negative. Concave → f negative. Convex → f positive. Real image → v negative. Virtual image → v positive.
Example: Concave mirror, u = 30 cm, f = 20 cm. Using signs: u = -30, f = -20. 1/v = 1/(-20) - 1/(-30) = -1/60; v = -60 cm (real). m = -(-60)/(-30) = -2 (inverted, 2x magnified).

📖 How to Use This Calculator

Steps

1
Select the mirror type and calculation mode: choose Find Image to compute where the image forms, or Find Focal Length if you know both object and image distances. Select Concave or Convex from the mirror type dropdown.
2
Enter object distance and focal length in centimetres (both as positive numbers). The calculator applies the correct New Cartesian sign convention internally based on your mirror type selection. In Find Focal Length mode, enter the image distance and select whether the image is real or virtual.
3
Optionally enter the object height in cm to also compute image height. You can leave this blank if only image position and magnification are needed.
4
Click Calculate to see image distance, image type (real or virtual), orientation (upright or inverted), magnification, image height, focal length, radius of curvature, and the full formula used with numerical values.

💡 Example Calculations

Example 1 — Concave Mirror, Object Beyond Center of Curvature

Concave mirror: object at 30 cm, focal length 20 cm, object height 2 cm

1
Apply New Cartesian signs: u = -30 cm, f = -20 cm (concave). Solve for v: 1/v = 1/f - 1/u = 1/(-20) - 1/(-30) = -0.05 + 0.0333 = -0.0167.
2
v = 1/(-0.0167) = -60 cm. Negative sign means real image, 60 cm in front of the mirror. Radius of curvature R = 2f = 40 cm.
3
Magnification: m = -v/u = -(-60)/(-30) = -2. Image is inverted and magnified 2 times. Image height = m times h_o = -2 times 2 = -4 cm (4 cm, inverted).
Image at 60 cm (real, inverted, magnified 2x) | Image height = 4 cm (inverted)
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Example 2 — Concave Mirror, Object Between Focus and Pole

Concave mirror (makeup mirror): object at 10 cm, focal length 20 cm

1
Signs: u = -10 cm, f = -20 cm. Solve: 1/v = 1/(-20) - 1/(-10) = -0.05 + 0.10 = +0.05.
2
v = 1/0.05 = +20 cm. Positive sign means virtual image, 20 cm behind the mirror. Object is inside the focal length, so the mirror acts as a magnifying mirror.
3
Magnification: m = -v/u = -(20)/(-10) = +2. Positive sign confirms upright image. Image is magnified 2 times.
Image at 20 cm behind mirror (virtual, upright, magnified 2x)
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Example 3 — Convex Mirror (Rear-View Mirror)

Convex mirror: object at 30 cm, focal length 15 cm

1
Signs: u = -30 cm, f = +15 cm (convex mirror has positive focal length). Solve: 1/v = 1/(15) - 1/(-30) = 0.0667 + 0.0333 = +0.10.
2
v = 1/0.10 = +10 cm. Virtual image, 10 cm behind the mirror (this is always the case for convex mirrors). Radius of curvature R = 2 times 15 = 30 cm.
3
Magnification: m = -(10)/(-30) = +0.333. Positive means upright. |m| = 0.333 means the image is diminished to one-third of the object size. Wide field of view is why convex mirrors are used as security mirrors and rear-view mirrors.
Image at 10 cm behind mirror (virtual, upright, diminished to 1/3)
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❓ Frequently Asked Questions

What is the mirror equation formula used in class 10 and 12 physics?+
The mirror equation is 1/v + 1/u = 1/f, where u is object distance, v is image distance, and f is focal length. Magnification is m = -v/u = h_i/h_o. For NCERT Class 10 and 12, the New Cartesian Sign Convention is used: real objects have u negative, concave mirrors have f negative, and real images have v negative. Virtual images have positive v.
How do I apply the New Cartesian sign convention for mirrors?+
In the New Cartesian Sign Convention, all distances are measured from the pole (vertex) of the mirror. Light travels left to right, so: (1) Object distance u is always negative for real objects placed in front of the mirror. (2) Focal length f is negative for concave mirrors, positive for convex. (3) Image distance v is negative for real images (in front) and positive for virtual images (behind). (4) Heights above the principal axis are positive; below are negative.
What is the difference between a real image and a virtual image in a mirror?+
A real image forms when reflected rays actually converge at a point in front of the mirror. It can be projected on a screen and is always inverted. A virtual image forms when reflected rays appear to diverge from a point behind the mirror. It cannot be projected on a screen and is always upright. Concave mirrors can produce either type depending on object position; convex mirrors always produce virtual images.
Why is the magnification negative for a concave mirror in some cases?+
Magnification m = -v/u. When the image is real (v negative) and the object is real (u negative), m = -(-v)/(-u) = -(v/u) which is negative if both v and u are negative with different magnitudes. A negative m means the image is inverted relative to the object. This happens for concave mirrors when the object is beyond the focal point, producing real inverted images.
What happens when the object is placed at the focus of a concave mirror?+
When u = f (object at the focal point), the denominator of the image distance equation becomes zero: 1/v = 1/f - 1/f = 0, giving v = infinity. This means the reflected rays are parallel and no image forms at a finite distance. In practice, this is how car headlights work: the bulb is placed at the focus to produce a parallel beam of light. This calculator shows an appropriate error message for this special case.
How do I find the focal length of a concave mirror at home?+
Hold the concave mirror facing a distant object (tree, building, far wall) so light from the object reflects and forms an image on a screen held in front. The distance from the mirror to the image is approximately the focal length (since the object distance is effectively infinite). More precisely, set up two known positions: measure u and v, then use the Find Focal Length mode to compute f = uv/(u+v).
What is the mirror formula for a convex mirror?+
The mirror formula is the same: 1/v + 1/u = 1/f. For a convex mirror, f is positive (virtual focus behind mirror). For a real object, u is negative. Solving always gives positive v (virtual image). Example: convex mirror with f = 15 cm, object at 30 cm: 1/v = 1/15 + 1/30 = 3/30 = 1/10; v = 10 cm. Image is virtual, upright, and diminished.
How is the image formed in a shaving or makeup mirror?+
A shaving or makeup mirror is a concave mirror. When you hold your face closer than the focal length, u is less than f. The mirror equation gives a positive v (virtual image), meaning the image appears to be behind the mirror. The magnification m = -v/u is positive and greater than 1, so the image is upright and magnified. This enlarged virtual image is what you see when using a magnifying makeup mirror.
What are the four characteristics used to describe an image in a mirror?+
The four standard characteristics are: (1) Position: where the image forms, measured from the mirror in cm. (2) Nature: real (can be projected on screen, forms in front of mirror) or virtual (cannot be projected, appears behind mirror). (3) Orientation: upright (same as object) or inverted (flipped). (4) Size: magnified (|m| greater than 1), diminished (|m| less than 1), or same size (|m| = 1). This calculator computes all four.
What is the relationship between focal length and radius of curvature for a mirror?+
For a spherical mirror, the focal length f equals half the radius of curvature R: f = R/2, or R = 2f. This relationship comes from the geometry of reflection: parallel rays striking a spherical mirror converge at the center between the mirror surface and the center of the sphere. A concave mirror with focal length 20 cm has its center of curvature at 40 cm, and a convex mirror with f = 15 cm has R = 30 cm.
What is linear magnification and how is it related to image and object size?+
Linear magnification m = h_i / h_o = -v/u, where h_i is image height and h_o is object height. If |m| is greater than 1, the image is magnified; if less than 1, it is diminished. A negative m means the image is inverted (negative height, below the principal axis). Enter the object height in this calculator to compute the actual image height in cm using h_i = m times h_o.
Tags: Mirror Equation Calculator Concave Mirror Calculator Convex Mirror Formula Mirror Magnification Calculator 1/v + 1/U = 1/F Spherical Mirror Calculator Calculator Formula Free

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What is the mirror equation and what does it solve?

The mirror equation is 1/v + 1/u = 1/f, where u is the object distance from the mirror, v is the image distance, and f is the focal length. It applies to both concave and convex spherical mirrors. Given any two of the three quantities, the equation solves for the third. Magnification is given by m = -v/u, and the radius of curvature R = 2f.

What sign convention does this mirror equation calculator use?

This calculator uses the New Cartesian Sign Convention, which is the standard for NCERT Class 10 and 12 physics. Object distances are always negative (real objects in front of mirror). Focal length of concave mirrors is negative; convex mirrors have positive focal length. Negative image distance means real image in front; positive means virtual image behind the mirror.

How do I find image distance using the mirror formula?

Enter the object distance and focal length in the Find Image mode, then select mirror type (concave or convex). The calculator computes 1/v = 1/f minus 1/u and returns the image distance. For a concave mirror with object at 30 cm and focal length 20 cm: 1/v = 1/(-20) - 1/(-30) = -1/60, giving v = -60 cm (real image, 60 cm in front).

What is the magnification formula for a mirror?

Magnification m = -v/u, where v is image distance and u is object distance (both with New Cartesian signs). A magnification of -2 means the image is inverted and twice as large. A magnification of +0.5 means the image is upright and half the size. For a concave mirror with u = -30 cm and v = -60 cm: m = -(-60)/(-30) = -2 (inverted, magnified 2 times).

Does a convex mirror always produce a virtual image?

Yes. A convex mirror always produces a virtual, upright, and diminished image regardless of where the object is placed, because its focal point is behind the mirror (positive focal length in Cartesian convention). The image always appears between the pole and the focus of the convex mirror. This is why convex mirrors are used as rear-view mirrors in vehicles.

What is the relationship between focal length and radius of curvature?

The radius of curvature R equals twice the focal length: R = 2f. For a concave mirror with focal length 20 cm, the center of curvature is at 40 cm from the mirror. This relationship holds for both concave and convex mirrors and follows directly from the geometry of reflection from a spherical surface.

When does a concave mirror produce a magnified image?

A concave mirror produces a magnified image when the object is placed between the focal point and the center of curvature (f to 2f). When the object is between the focus and the pole, the image is virtual, upright, and magnified (|m| > 1). This is the principle behind makeup mirrors and shaving mirrors. When the object is beyond 2f, the image is real, inverted, and diminished.

How do I use the mirror equation to find focal length?

Use the Find Focal Length mode. Enter the object distance and image distance, specify whether the image is real (in front) or virtual (behind), and click Calculate. The calculator solves 1/f = 1/v + 1/u. For object at 30 cm and real image at 60 cm: 1/f = 1/(-60) + 1/(-30) = -1/20, giving f = -20 cm (concave, focal length 20 cm).

What are the uses of a concave mirror in real life?

Concave mirrors are used as: shaving and makeup mirrors (object inside focal length gives magnified upright image), dental mirrors, headlight reflectors in cars and flashlights (object at focus gives parallel beam), solar concentrators (focusing sunlight to heat), reflecting telescopes (primary mirror), and satellite dish feeds. They can form both real and virtual images depending on object position.

What does it mean when image distance is infinity in the mirror formula?

When the object is placed exactly at the focal point of a concave mirror, 1/v = 1/f - 1/u = 0, so v = infinity. This means the reflected rays are parallel and never converge to form an image at a finite distance. This is used in car headlights and searchlights, where the bulb is placed at the focus to produce a parallel beam of light.

📌 Quick Tips

💡For a concave mirror, the focal length is negative in the New Cartesian sign convention. This calculator accepts the absolute value (positive number) and applies the correct sign internally based on mirror type.
💡A real image forms in front of the mirror and can be projected on a screen. A virtual image forms behind the mirror and cannot be projected. Convex mirrors always produce virtual images.
💡Magnification m = -v/u. A negative m means the image is inverted (real image for concave mirrors). A positive m means the image is upright (virtual images are always upright in spherical mirrors).
💡The radius of curvature R = 2f. A concave mirror with focal length 20 cm has a center of curvature at 40 cm. Objects placed beyond 40 cm produce diminished real images.
💡For NCERT Class 10 and 12 physics, the sign convention used here is the New Cartesian Sign Convention where all distances are measured from the pole of the mirror and distances opposite to incident light are negative.