Mirror Equation Calculator

Mirror Equation Calculator that solves 1/f = 1/v + 1/u for image distance, focal length or object distance, then shows magnification, image nature and a ray diagram for any mirror.

Using the Mirror Equation Calculator

The Mirror Equation Calculator solves the mirror formula for whichever value you are missing. It finds the image distance from the object distance and focal length, the focal length from the object distance and either the image distance or the magnification, the object distance from an image distance, or the object position that gives a chosen magnification.

Pick a concave, convex or plane mirror, and choose New Cartesian signs (u, v) or the real-is-positive signs (dₒ, dᵢ) used in many US textbooks. Every answer comes with the magnification and image height, the nature of the image, the radius of curvature, a ray diagram and the step-by-step math.

The Mirror Formula

The mirror formula links the object distance u, the image distance v and the focal length f of a spherical mirror. The NCERT Class 10 chapter on reflection and refraction states that it holds for all spherical mirrors and all positions of the object.

$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

The focal length is half the radius of curvature, f = R/2, so a mirror with R = 24 cm has f = 12 cm. Card 3 of the Mirror Equation Calculator shows R from f, or f from R if you switch Mirror Given By to the radius.

What u, v and f Mean

All three distances are measured from the pole P, the center of the mirror’s surface. In the New Cartesian convention used by NCERT, distances along the incoming light are positive and distances against it are negative, with the object always placed on the left.

So v is negative for a real image formed by a concave mirror, because that image sits in front of the mirror on the object’s side. A convex mirror’s focal length is positive, while a concave mirror’s is negative. If you type a sign that does not fit the mirror you picked, the calculator says so and gives the value to enter instead.

How to Solve the Mirror Equation

Write the known values with their signs, put them into the formula, and solve for the unknown. Take the calculator’s default case, a concave mirror with f = −12 cm and an object 9 cm in front, so u = −9 cm.

$$\frac{1}{v} = \frac{1}{f} – \frac{1}{u} = \frac{1}{-12} – \frac{1}{-9} = \frac{1}{36}$$

So v = +36 cm. The positive sign puts the image 36 cm behind the mirror, which makes it virtual. This is the shaving mirror case, where the face sits inside the focus and looks four times larger.

The same steps work for any unknown. For the focal length, add 1/v and 1/u and take the reciprocal. If you know the magnification instead of v, find v = −m × u first. For the object distance, subtract 1/v from 1/f. The calculator does this in its four modes and checks that the result can come from a real object in front of the mirror.

Real-Is-Positive Signs

Many US physics books write the same law with object distance dₒ and image distance dᵢ, and count real objects and real images as positive. Concave mirrors then have a positive focal length, and convex mirrors a negative one.

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$

In these signs the default case is dₒ = 9 cm and f = +12 cm, which gives dᵢ = −36 cm, a virtual image. Card 4 of the Mirror Equation Calculator translates each answer into the other convention, so a result can be checked against either kind of textbook.

Mirror Magnification

Linear magnification compares the image height with the object height. For a mirror it equals minus the ratio of the image and object distances.

$$m = \frac{h_i}{h_o} = -\frac{v}{u}$$

A negative m means a real, inverted image, and a positive m means a virtual, upright one. In the default case m = −(36) ÷ (−9) = +4, so a 2 cm object gives an 8 cm upright image. Card 1 also shows the areal magnification m², which is 16 here, the factor by which the image area grows.

Magnification is not always −v/u. For a thin lens, NCERT writes m = h′/h = v/u without the minus sign, because the lens formula uses a different sign pattern. The minus sign in the mirror version comes from the image and object lying on the same side of the mirror for a real image.

The Position for a Magnification mode works backward from m. Combining m = −v/u with the mirror formula gives the object distance for any target magnification.

$$u = f\left(1 – \frac{1}{m}\right)$$

For a concave mirror with f = −12 cm, an inverted image twice the object’s size (m = −2) needs u = −18 cm, and the image forms at v = −36 cm.

Concave Mirror Equation

A concave mirror curves inward like the inside of a spoon and converges light. Its image changes with the object’s position, from real and inverted far away to virtual and upright inside the focus, so it is the only mirror that can form an inverted image.

The table runs a concave mirror with f = −12 cm (R = −24 cm) through the object positions in NCERT’s image-formation table, from beyond C to inside F. The Mirror Equation Calculator highlights the matching row for your own object in its Concave Mirror Image Rules table.

An object at infinity, such as sunlight, forms a point image at F. Placing a bulb at F runs this in reverse, which is why NCERT lists torches, searchlights and vehicle headlights among the uses of concave mirrors. Shaving and dental mirrors use the virtual, enlarged image, and solar furnaces use large concave mirrors to concentrate sunlight.

Ray Rules for Spherical Mirrors

A ray diagram needs only two of four standard rays. NCERT gives them in this order, and the third rule is the one about the center of curvature.

A ray through C meets the mirror along a normal, so it comes straight back. The calculator’s Ray Diagram uses the parallel ray and the ray to the pole, and it draws virtual rays as dashed lines behind the mirror.

Convex Mirror Equation

A convex mirror bulges outward and spreads light, so its image is always virtual, upright and smaller than the object. The image always forms between P and F behind the mirror, and it moves toward F as the object moves away.

The values above use f = +12 cm. The four ray rules apply to a convex mirror too, with the reflected rays traced backward to meet behind it. That wide, shrunken view is why convex mirrors serve as rear-view mirrors on vehicles and as security mirrors in shops.

NCERT’s Class 10 Example 9.1 works a bus mirror with R = +3.00 m and a car 5.00 m away. The Mirror Equation Calculator gives v = +1.15 m and m = +0.23, so the car looks about a quarter of its size, 1.15 m behind the mirror.

Plane Mirror Image

A plane mirror is flat, so its radius of curvature and focal length are infinite and 1/f becomes zero. The mirror formula then reduces to v = −u, which puts the image as far behind the mirror as the object is in front.

The image is virtual, upright and the same size, so m = +1. A person 1.5 m in front of a wall mirror sees an image 1.5 m behind it, 3 m from their eyes. The Position for a Magnification mode does not accept a plane mirror, since no object position can change m = +1.

A plane mirror flips the scene front to back, which makes printed text read backward. That is why your mirror image is not how others see you. A non-reversing mirror joins two mirrors at a right angle so the two reflections cancel the swap and show the face the way other people see it.

Newton’s Mirror Formula

Newton’s form of the mirror equation measures distances from the focus instead of the pole. If xₒ is the object’s distance from F and xᵢ is the image’s distance from F, their product equals the square of the focal length.

$$x_o \cdot x_i = f^2$$

Card 3 of the Mirror Equation Calculator lists both distances. In the default case xₒ = 3 cm and xᵢ = 48 cm, and 3 × 48 = 144 = 12². Every row in the concave table above passes the same check, since f is 12 cm in each case.

Mirror Equation Derivation

The NCERT Class 12 Ray Optics chapter derives the mirror equation in Section 9.2.3 from two pairs of similar triangles. Let AB be the object, A′B′ its image, and M the point where the parallel ray from A meets the mirror, with PM taken as equal to AB for rays close to the axis.

Triangles A′B′F and MPF are similar, and so are triangles A′B′P and ABP. Each pair gives a ratio for the image height over the object height.

$$\frac{A’B’}{AB} = \frac{B’F}{FP} = \frac{B’P}{BP}$$

Write B′F as B′P − FP, then put in the signed distances for a concave mirror. In New Cartesian signs B′P = −v, FP = −f and BP = −u, because all three points lie in front of the mirror.

$$\frac{-v + f}{-f} = \frac{-v}{-u}$$

Cancelling the minus signs leaves (v − f)/f = v/u. Dividing every term by v gives 1/f − 1/v = 1/u, which rearranges to the mirror equation that the Mirror Equation Calculator solves.

$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

The second pair of triangles also gives the magnification, since A′B′/AB = B′P/BP becomes −h′/h = −v/−u, or m = −v/u. NCERT numbers these results Eq. 9.7 and Eq. 9.9.

Mirror Formula in Class 10 and 12

In the CBSE syllabus, the mirror formula appears at both levels. The CBSE Class 10 Science syllabus for 2025–26 lists the mirror formula and the lens formula with “derivation not required”, along with magnification.

The CBSE Class 12 Physics syllabus for 2025–26 lists spherical mirrors and the mirror formula in Chapter 9, Ray Optics, without that note. Class 12 students should therefore know the derivation above, while Class 10 students need the formula, the signs and the magnification.

A common Class 10 exercise asks where the image forms when a concave mirror makes a real image three times the size of an object 10 cm in front of it. A real image means m = −3, so v = −m × u = −(−3)(−10) = −30 cm, and the mirror formula then gives f = −7.5 cm.

The Focal Length mode of the Mirror Equation Calculator takes this case directly. Enter u = −10 cm, set Second Known Value to Magnification (m) and type −3. It returns f = −7.5 cm and R = −15 cm for a concave mirror, with the image at v = −30 cm beyond C and nine times the object’s area.

NCERT’s Class 10 Example 9.2 uses a 4.0 cm object 25.0 cm from a concave mirror with f = −15.0 cm. The calculator gives v = −37.5 cm and a −6.0 cm image, real, inverted and enlarged, so a screen 37.5 cm in front of the mirror catches it.

Mirror Formula and Lens Formula

Students often meet the two formulas side by side. They look alike, but the lens formula has a minus sign because a lens forms real images on the far side, where v is positive.

$$\frac{1}{f} = \frac{1}{v} – \frac{1}{u}$$

So f is positive for a convex lens and negative for a concave lens, and a convex lens gives a positive v whenever its image is real. Convex lenses come as biconvex, plano-convex and concavo-convex (positive meniscus) shapes, and the Lens Maker Equation Calculator finds f for each from its radii and glass index. The Diopter Calculator turns any focal length into lens power in dioptres.

Mirror Equation Worksheet

These practice problems work as a short mirror solver drill and cover every mode of the calculator, with answers in New Cartesian signs. Enter the givens in the Mirror Equation Calculator to see each step, the image nature and the ray diagram.

Problem 6 matches NCERT’s Class 12 Example 9.3, where the same mirror gives a real image at −30 cm for an object at −10 cm and a virtual image at +15 cm for an object at −5 cm. In both cases the size of the image is three times the object.

Mirror Formula Uses

The mirror formula finds where an image forms and how large it is without drawing a ray diagram. Opticians, telescope makers and lighting designers use it to place a screen, a sensor or a bulb at the right distance from a curved mirror.

A reflecting telescope collects light with a large concave mirror, and its resolving limit comes from the mirror’s diameter, as the Angular Resolution Calculator shows. A headlight reflector puts its bulb at F to throw a parallel beam, and the Lumen Calculator works out how far a beam of that kind reaches.

The letters u and v also appear in the equations of motion, where they mean initial and final velocity rather than distances. Those formulas, such as v = u + at, belong to kinematics, and the SUVAT Calculator solves them. More optics tools sit in the Optics Calculators hub.