Power of Lens Formula: P = 1/f, Dioptres and Solved Examples

The power of lens formula is $P = \frac{1}{f}$ , where f is the focal length in metres. The answer comes out in dioptres (D). A lens with a focal length of 0.5 m has a power of 2 D, and one with a focal length of 0.25 m has a power of 4 D. The shorter the focal length, the more strongly the lens bends light and the larger its power.

Opticians write spectacle prescriptions in dioptres because powers of thin lenses in contact simply add. A sign comes with every power. In the New Cartesian sign convention used in Class 10 and Class 12, a convex (converging) lens gets a positive f and so a positive P, while a concave (diverging) lens has a negative focal length and a negative power.

Quick facts

Formula
$P = \frac{1}{f}$ with f in metres
With f in cm
$P = \frac{100}{f}$
SI unit
dioptre (D), where 1 D = 1 m⁻¹
Dimensional formula
[M⁰ L⁻¹ T⁰]
Sign
Convex lens: positive. Concave lens: negative
Lenses in contact
$P = P_1 + P_2 + P_3 + \dots$
Used in
Spectacles, cameras, microscopes, Class 10 human eye chapter

Power of lens formula, term by term

Terms in P = 1 / f
Symbol	Meaning	SI unit
$P$	Power of the lens, its ability to converge or diverge light	D (m⁻¹)
$f$	Focal length, from optical centre to principal focus	m

Power is the reciprocal of focal length, so it measures how quickly a lens brings a parallel beam to a point. A strong magnifying lens has a short focal length and a high power. A nearly flat lens has a long focal length and a power close to zero. A plane sheet of glass has an infinite focal length and zero power.

The formula only gives dioptres if f is in metres. If the question gives centimetres, either convert first or use the shortcut $P = \frac{100}{f(\text{cm})}$ .

Using the formula step by step

Identify the lens. Convex lenses get a positive f, concave lenses a negative f.
Convert the focal length to metres. 25 cm becomes 0.25 m.
Take the reciprocal, keeping the sign: $P = \frac{1}{f}$ .
Write the unit D and the sign in the answer. A prescription of −2.5 D tells the optician to supply a diverging lens.

To go backward from a power to a focal length, use $f = \frac{1}{P}$ and then multiply by 100 if you want centimetres. The calculator on this page does both directions in SI units.

Worked examples

Example 1: a reading lens

A convex lens has a focal length of 25 cm. Find its power.

Given: f = +25 cm = +0.25 m.

$P = \frac{1}{0.25} = +4.0\ \text{D}$

Example 2: a spectacle lens

A concave lens has a focal length of 50 cm. What is its power?

Given: f = −50 cm = −0.50 m.

$P = \frac{1}{-0.50} = -2.0\ \text{D}$

The minus sign marks a diverging lens, the type used to correct short sight (myopia).

Example 3: two lenses in contact

A thin convex lens of power +4.0 D is placed in contact with a thin concave lens of power −1.5 D. Find the power and focal length of the combination.

$P = P_1 + P_2 = 4.0 + (-1.5) = +2.5\ \text{D}$

$f = \frac{1}{P} = \frac{1}{2.5} = 0.40\ \text{m} = 40\ \text{cm}$

The pair acts as one converging lens of focal length 40 cm.

Example 4: a prescription from a far point

A short sighted student cannot see clearly beyond 2.0 m. What lens power does she need?

The lens must make distant objects appear to sit at her far point, so f = −2.0 m.

$P = \frac{1}{-2.0} = -0.50\ \text{D}$

Unit and dimensions of power of a lens

Power is one divided by a length, so it carries the dimensions [M⁰ L⁻¹ T⁰] and the SI unit is m⁻¹. That unit has the name dioptre. One dioptre is the power of a lens whose focal length is exactly one metre.

Do not confuse this with power in mechanics, measured in watts. The two share a name only. Lens power has nothing to do with energy per second.

Power and focal length side by side
Focal length	Power	Lens type
+1.0 m	+1 D	Weak converging
+0.10 m	+10 D	Strong converging, like a hand magnifier
−0.40 m	−2.5 D	Diverging
Infinite (flat glass)	0 D	No focusing

How lens power connects to other optics formulas

The focal length that goes into P = 1/f can come from the lens maker formula, which builds the power straight from the refractive index and the two radii of curvature. Once you know the power, the lens formula locates the image, and the magnification of lens formula sizes it.

A lens with one face coated to act as a mirror is handled with powers too: the silvering of lens formula counts the lens power twice because light passes through it twice. Lens power is also a key idea in the eye, where the eye lens changes its power to focus near and far objects. More optics formulas sit on the waves and optics hub.

Where students lose marks

Using centimetres. 1/25 = 0.04 is not a power in dioptres. Convert 25 cm to 0.25 m first.
Dropping the sign. −2 D and +2 D are completely different lenses.
Adding focal lengths. For lenses in contact, add powers, not focal lengths. Convert back to f at the end.
Ignoring the surrounding medium. The quoted power of a lens assumes it sits in air. Dip the same glass lens in water and its focal length grows, so its power drops.

A higher number of dioptres, plus or minus, usually means a more strongly curved lens.

Frequently asked questions

What is the formula of power of lens?

Power is P = 1/f, where f is the focal length in metres. With f in centimetres you can use P = 100/f.

What is the SI unit of power of a lens?

The dioptre, symbol D. It equals one per metre, so a lens of focal length 1 m has a power of 1 D.

Why is the power of a concave lens negative?

A concave lens diverges light and its focal length is negative in the New Cartesian sign convention. The reciprocal of a negative number is negative, so the power is negative.

How do you find the power of two lenses in contact?

Add the individual powers with their signs: P = P₁ + P₂. The focal length of the pair is then 1/P.

What does a power of +2.5 D mean on a prescription?

It is a converging lens with a focal length of 1 ÷ 2.5 = 0.40 m, or 40 cm. Such lenses are used to correct long sight (hypermetropia) and reading difficulty.