Lens & Mirror Equation Calculator

Optics Guide

What should I know about The Thin Lens Equation?

1/f = 1/v + 1/u. Where f = focal length, v = image distance, u = object distance. Sign convention (real-is-positive): real object (in front of lens): u positive. Real image (on opposite side from object): v positive. Virtual image (same side as object): v negative. Converging lens: f positive. Diverging lens: f negative. Lens power: P = 1/f(metres) in dioptres (D). A 25 cm focal length lens: P = 1/0.25 = +4D. Short-sighted (myopia) is corrected with diverging lenses (negative power). Long-sightedness, or hyperopia, is corrected with converging lenses (positive power) instead, since the eye focuses light behind rather than on the retina.

What should I know about Image Characteristics?

To determine image type: real image: rays actually pass through the image point — can be projected on a screen — formed on the opposite side of the lens from the object. Virtual image: rays appear to diverge from the image point — cannot be projected — formed on the same side as the object. Erect vs inverted: m > 0: erect (same way up as object). m < 0: inverted. Magnified vs diminished: |m| > 1: larger than object. |m| < 1: smaller. |m| = 1: same size. For a converging lens: an object placed beyond twice the focal length produces a real, inverted, and diminished image — the configuration used in cameras.

What should I know about Mirror Equation?

Same formula applies to mirrors: 1/f = 1/v + 1/u. Concave (converging) mirror: f = R/2 (where R = radius of curvature), f positive. Convex (diverging) mirror: f negative. Sign convention for mirrors: real object (in front of mirror): u positive. Real image (in front of mirror, same side as object for reflection): v positive. Virtual image (behind mirror): v negative. A concave mirror forms a real, inverted image when the object is beyond the focal point — this is how reflecting telescopes and satellite dishes both work, using a curved mirror to focus incoming light or signals to a single point.

What should I know about Applications?

Camera: converging lens forms a real, inverted image on a sensor. Projector: object (slide/LCD) just beyond f — real, inverted, greatly magnified image on screen. Magnifying glass: object within f — virtual, erect, magnified image. Telescope: objective lens forms real image in focal plane; eyepiece acts as magnifying glass for this intermediate image. Eye: cornea (main converging power ~43D) and crystalline lens (variable 10-20D) form a real, inverted image on the retina. Total eye power is approximately 60 dioptres, combining both the fixed corneal power and the variable lens power the eye adjusts to focus at different distances.

What should I know about Lens Power and Diopters?

Optometrists and opticians describe lens strength as power P, measured in diopters (D), where P = 1/f with f in metres. A converging lens with f = 0.5 m has a power of +2 D; a diverging lens with f = −0.25 m has a power of −4 D. Diopters are useful because, unlike focal length, they add directly: two thin lenses placed in contact combine to a total power P_total = P₁ + P₂, which is exactly how opticians stack trial lenses during an eye exam to find a prescription.

What do I need to know about Combinations of Two Lenses?

When two thin lenses sit close together, their powers simply add: 1/f_total = 1/f₁ + 1/f₂. This is how compound microscopes and telescopes work — an objective lens forms a first image, which then becomes the object for an eyepiece lens that magnifies it further, multiplying the individual magnifications together for a much larger overall magnification than either lens could achieve alone. For lenses separated by a distance d, the simple power-addition rule breaks down and the full two-lens system equation (accounting for d) must be used instead.

What should I know about Sign Conventions in the Lens Equation?

The thin lens equation 1/f = 1/v − 1/u (or 1/u + 1/v = 1/f, depending on convention) only gives correct answers when every distance carries the right sign, and mixing conventions is the most common source of errors. Under the widely used 'real-is-positive' convention, object distances are taken as positive when measured against the direction of incoming light, converging (convex) lenses have positive focal length, and diverging (concave) lenses have negative focal length. A positive image distance v means the image forms on the far side of the lens from the object and is real (it can be projected onto a screen); a negative v means the image is virtual, forming on the same side as the object. Magnification, m = −v/u, then follows the same sign logic: a negative m indicates an inverted image, a positive m an upright one. Because textbooks and calculators sometimes use the alternative 'Cartesian' convention, where all distances are measured from the lens with a consistent positive direction, it is important to check which convention a given formula assumes before plugging in numbers, since swapping conventions silently flips the sign of every answer.

What Focal Lengths Appear in Common Real-World Optical Instruments?

Focal length determines how strongly a lens bends light, and seeing the range used in familiar devices makes the abstract number from the thin lens equation easier to sanity-check.

Worked example: a magnifying glass with a 10cm focal length held so an object sits 7cm away (inside the focal length, producing a magnified virtual image) can be checked with the thin lens equation: 1/f = 1/do + 1/di → 1/10 = 1/7 + 1/di → 1/di = 0.1 - 0.1429 = -0.0429 → di ≈ -23.3cm. The negative image distance confirms the image is virtual (on the same side as the object, not projectable onto a screen) — exactly what's expected for a magnifying glass used in its normal close-up viewing mode, where the object sits inside the focal length.

What's the Difference Between a Converging and a Diverging Lens in the Equation?

The thin lens equation (1/f = 1/do + 1/di) uses a sign convention where a converging (convex) lens has a positive focal length and a diverging (concave) lens has a negative focal length — this single sign difference is what produces fundamentally different image behavior between the two lens types. A converging lens can produce either a real, inverted image (when the object is beyond the focal length) or a magnified virtual image (when the object is inside the focal length, as with a magnifying glass). A diverging lens, by contrast, always produces a virtual, upright, reduced image regardless of where the object is placed, which is exactly why diverging lenses are used in applications like myopia-correcting eyeglasses, where a smaller, upright virtual image is the desired correction rather than the real, invertible image a converging lens would otherwise produce.

Worked Example: Image Position and Magnification for a Converging Lens

An object is placed 30 cm in front of a converging lens with a focal length of 10 cm. Using the thin lens equation 1/f = 1/v - 1/u (with the sign convention where distances are measured from the lens, object distance u is negative for a real object), or more simply using 1/f = 1/v + 1/u with both u and f positive for a real object and converging lens: 1/10 = 1/v + 1/30, so 1/v = 1/10 - 1/30 = 3/30 - 1/30 = 2/30 = 1/15, giving v = 15 cm. The image forms 15 cm on the far side of the lens. Magnification is m = -v/u = -15/30 = -0.5, meaning the image is inverted (negative sign) and half the size of the object (magnitude 0.5). This is the configuration used in a basic magnifying arrangement for projecting a reduced, inverted real image, such as in a simple camera lens forming an image on a sensor smaller than the object being photographed.

Human Eye as an Optical System

The human eye functions as a single converging-lens system where the retina acts as the fixed image screen, roughly 2.4 cm behind the cornea and lens. Because the image distance is fixed by the eye's physical shape, the eye cannot move its 'screen' the way a camera moves its sensor back and forth to focus -- instead, muscles change the lens's curvature (and therefore its focal length) to keep images focused on the retina regardless of object distance, a process called accommodation. A healthy young eye can adjust its effective focal length roughly between 17 mm (focusing on something close) and slightly more (focusing on something distant), but this range of accommodation decreases with age, which is why presbyopia (the common age-related loss of near-focusing ability) typically sets in during a person's 40s, and why corrective lenses for myopia, hyperopia, and presbyopia are themselves applications of the same thin lens equation covered in this calculator.