Mirror Equation Calculator - Calculate Focal Length & Magnification

What is the Mirror Equation?

Core Formula
Key Variables
Sign Conventions

The mirror equation is a fundamental formula in optics that relates the object distance (do), image distance (di), and focal length (f) of a spherical mirror. It is expressed as: 1/f = 1/do + 1/di. This equation is essential for determining where an image will be formed by a concave or convex mirror and its characteristics.

Understanding the Components

Object Distance (do): The distance from the object to the center (vertex) of the mirror. It is almost always a positive value. Image Distance (di): The distance from the formed image to the mirror's vertex. A positive value indicates a real image (formed on the same side as the object), while a negative value signifies a virtual image (formed behind the mirror). Focal Length (f): The distance from the mirror's vertex to its focal point. By convention, it is positive for concave (converging) mirrors and negative for convex (diverging) mirrors.

Sign Convention Rules

f is positive (+) for a concave mirror.
f is negative (-) for a convex mirror.
di is positive (+) for a real image, located in front of the mirror.
di is negative (-) for a virtual image, located behind the mirror.
ho (object height) is positive if upright; hi (image height) is negative if inverted.

Step-by-Step Guide to Using the Calculator

Selecting the Goal
Inputting Values
Interpreting Results

Our calculator simplifies the process. First, choose the variable you wish to calculate (Image Distance, Object Distance, or Focal Length). Next, select the mirror type (Concave or Convex), which automatically handles the focal length's sign. Fill in the known values, including an optional object height for magnification calculations. Finally, click 'Calculate' to see the result, along with magnification and a description of the image's properties (real/virtual, upright/inverted).

Example Calculation

Goal: Find Image Distance (di).
Inputs: Concave Mirror, Object Distance (do) = 30 cm, Focal Length (f) = 15 cm.
Calculation: 1/15 = 1/30 + 1/di => 1/di = 1/15 - 1/30 = 1/30 => di = 30 cm.
Result: The image is real, located 30 cm from the mirror.

Real-World Applications of Spherical Mirrors

Concave Mirrors
Convex Mirrors
Technological Uses

Concave mirrors are used in applications that require magnification or focusing light, such as shaving/makeup mirrors, dental instruments, and headlights in cars. They can form both real and virtual images. Convex mirrors, which provide a wider field of view, are commonly used for security purposes in stores and as side-view mirrors on vehicles ('Objects in mirror are closer than they appear').

Everyday Examples

Car side-view mirrors use convex mirrors for a wider view.
Satellite dishes use a large concave reflector to focus signals onto a receiver.
Telescopes like the Hubble use large concave mirrors to gather light from distant stars.

Magnification and Image Properties

Calculating Magnification
Real vs. Virtual Images
Upright vs. Inverted Images

Magnification (m) tells us how large the image is relative to the object and its orientation. It's calculated as m = -di / do. A negative magnification means the image is inverted, while a positive value means it is upright. A magnification with an absolute value greater than 1 means the image is larger than the object. Real images are formed where light rays actually converge and can be projected onto a screen. Virtual images are formed where light rays appear to diverge from and cannot be projected.

Interpreting Magnification

If m = -2.0, the image is real (implied by positive di), inverted, and twice the size of the object.
If m = +0.5, the image is virtual (implied by negative di), upright, and half the size of the object.

Mathematical Derivation and Edge Cases

Geometric Optics Derivation
Object at Focal Point
Object at Center of Curvature

The mirror equation is derived using similar triangles from a ray diagram. An interesting edge case occurs when an object is placed at the focal point (do = f) of a concave mirror. The equation becomes 1/f = 1/f + 1/di, leading to 1/di = 0. This means the image distance is at infinity, and the outgoing rays are parallel. When the object is at the center of curvature (do = 2f), the image is also formed at the center of curvature (di = 2f), resulting in a real, inverted image of the same size (m = -1).

Special Cases for Concave Mirrors

Object at F: Image is formed at infinity.
Object at C (2F): Image is formed at C, real, inverted, same size.
Object between C and F: Image is formed beyond C, real, inverted, magnified.

Concave Mirror: Real Image

Concave Mirror: Virtual Image

Convex Mirror: Virtual Image

Finding Focal Length