Lens and Mirror Equation Calculator

The lens and mirror equation

For a thin lens and for a spherical mirror alike, the object distance, the image distance and the focal length obey the same relation:

1 / f = 1 / dₒ + 1 / dᵢ

f is the focal length, dₒ the distance from the object to the lens or mirror and dᵢ the distance to the image. For a spherical mirror the focal length is half the radius of curvature: f = R / 2.

Magnification

m = −dᵢ / dₒ

The absolute value of the magnification tells you how many times larger the image is than the object, and its sign tells you whether the image is upright or inverted. The image height is hᵢ = m × hₒ.

Sign convention

• The focal length is positive for a converging (convex) lens and a concave mirror, negative for a diverging (concave) lens and a convex mirror.
• A positive dᵢ means a real image that can be caught on a screen. A negative dᵢ means a virtual image.
• A positive m means an upright image and a negative m an inverted one.

Example

For an object 50 cm in front of a converging lens with a focal length of 10 cm, 1 / dᵢ = 1/10 − 1/50 = 4/50, so dᵢ = 12.5 cm. The magnification is −12.5 / 50 = −0.25: the image is real, inverted and a quarter of the size of the object. Move the same object to 5 cm from the lens, inside the focal point, and dᵢ = −10 cm with m = +2: the image is virtual, upright and twice as large. That is how a magnifying glass works.

Special cases

With the object exactly at the focal point the rays leave parallel and the image forms at infinity. A diverging lens or a convex mirror always gives a virtual, upright and smaller image of a real object.

Limits

The equation is for thin lenses and rays close to the axis. It is only approximate for thick lenses, lens systems and mirrors with a wide aperture.