Concave and Convex Mirrors: Ray Diagrams and Equation

A concave mirror curves inward and converges parallel rays to a real focal point at f = R ÷ 2, while a convex mirror curves outward and diverges rays so that they appear to come from a virtual focal point behind it; the mirror equation 1/f = 1/do + 1/di and the magnification m = −di ÷ do are the same as for thin lenses. In brief: a concave mirror can form real, inverted images (object beyond f) or a virtual, magnified, upright image (object inside f, as in a shaving mirror). A convex mirror always forms a virtual, upright, diminished image over a wide field of view, which is why it is used as a rear-view or security mirror.

Telescope mirrors, car headlights, makeup mirrors and shop security mirrors are curved mirrors. This guide describes the geometry of spherical mirrors, the three principal rays, the mirror equation and sign conventions, compares the mirror types in a table, works through numerical examples and corrects common mistakes.

What are concave and convex mirrors?

A spherical mirror is part of a reflecting sphere of radius R. The center of the sphere is the center of curvature C, and the line through C and the middle of the mirror (the vertex) is the principal axis. For rays close to the axis, parallel rays are reflected through (concave) or away from (convex) a single point, the focal point F, at a distance f = R ÷ 2 from the mirror. Wider beams do not focus perfectly (spherical aberration); telescope mirrors and satellite dishes use a parabolic shape, which focuses parallel rays exactly.

How are ray diagrams drawn?

  1. A ray parallel to the axis is reflected through F (concave) or appears to come from F behind the mirror (convex).
  2. A ray through F (concave; or directed toward F behind the convex mirror) is reflected parallel to the axis.
  3. A ray through C (or directed toward C) strikes the mirror along the normal and is reflected back along itself.
  4. A ray to the vertex is reflected at the same angle on the other side of the axis.

Where two reflected rays meet is the image of the top of the object. If they only appear to meet behind the mirror, the image is virtual.

What is the mirror equation?

The mirror equation 1/f = 1/do + 1/di and the magnification m = −di ÷ do have the same form as for thin lenses. The sign convention: do is positive for a real object in front of the mirror; di is positive for a real image in front of the mirror and negative for a virtual image behind it; f is positive for concave and negative for convex mirrors; m > 0 for upright and m < 0 for inverted images. For a plane mirror, f → ∞ and di = −do, with m = +1.

Ray diagram of a concave mirror forming a real inverted image C F object image R = 60 cm, f = 30 cm, do = 90 cm → di = 45 cm
Concave mirror with R = 60 cm (f = 30 cm) and an object 90 cm away, drawn as a straight line as is usual for rays near the axis. The ray parallel to the axis reflects through F, and the ray aimed through F reflects parallel to the axis. They meet 45 cm from the mirror: the image is real, inverted and half the size (m = −0.50).

Concave and convex mirrors compared

Images formed by spherical mirrors
Mirror and object position Image Use
Concave, beyond CReal, inverted, diminishedTelescope mirror (distant object)
Concave, between C and FReal, inverted, magnifiedProjector, headlight source
Concave, at FImage at infinity (parallel beam)Torch, car headlights, searchlights
Concave, inside FVirtual, upright, magnifiedShaving or makeup mirror
Convex, any positionVirtual, upright, diminishedRear-view and security mirrors

Worked example: image calculations

  1. Concave mirror, object far. A concave mirror has R = 60 cm, so f = 30 cm. An object is placed 90 cm in front. 1/di = 1/30 − 1/90 = 0.0333 − 0.0111 = 0.0222, di = 45 cm. m = −45 ÷ 90 = −0.50: real, inverted, half the size.
    F C 90 cm 45 cm
  2. Shaving mirror. The same mirror, with the object (a face) 15 cm away: 1/di = 1/30 − 1/15 = −0.0333, di = −30 cm (virtual, 30 cm behind the mirror). m = −(−30) ÷ 15 = +2.0: upright and twice as large.
  3. Convex mirror. A convex mirror with f = −40 cm, and a car 120 cm away: 1/di = −1/40 − 1/120 = −0.0250 − 0.00833 = −0.0333, so di = −30 cm. m = +30 ÷ 120 = +0.25. The image is virtual, upright and a quarter of the size, which lets the driver see a wide field.
  4. Parabolic reflector. A torch bulb is placed at the focus of a concave mirror, and the reflected rays leave parallel to the axis, forming a beam.
  5. Sun’s image. The Sun’s rays are parallel, and a concave mirror with f = 30 cm focuses them to a spot at 30 cm, hot enough to ignite paper (a solar furnace uses large concave mirrors).

How were curved mirrors used?

According to legend, Archimedes used mirrors to set Roman ships on fire at Syracuse in 212 BCE, though this is doubtful. Ibn al-Haytham studied spherical mirrors and their aberrations around 1021. Isaac Newton built the first practical reflecting telescope with a concave mirror in 1668, avoiding the chromatic aberration of lenses. Today the largest optical telescopes, such as the 10 m Keck telescopes, and the James Webb Space Telescope (6.5 m), use segmented mirrors.

Which mistakes are common?

Mistake 1: Using f = R instead of f = R ÷ 2

The focal point is halfway between the mirror and the center of curvature.

Mistake 2: Putting the convex mirror’s focal point in front of the mirror

It is behind the mirror, so f is negative.

Mistake 3: Saying that concave mirrors always magnify

Beyond the center of curvature, the image is smaller than the object.

Mistake 4: Confusing the signs of mirror and lens image distances

For mirrors, real images are in front of the mirror, on the same side as the object. For lenses, real images are on the opposite side.