Magnification Calculator
Calculate magnification calculator calculate optical linear magnification M (-di / do) and image height hi from object height ho and distances (M = -di/do = hi/ho). Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Optical Magnification (M = -d_i / d_o = h_i / h_o)
Enter your physical parameters below to compute verified magnification metrics.
Calculation Results
Calculated using verified physical methodology: Linear Magnification: M = -\d_i / d_o
Image Height: h_i = M \cdot h_o = -h_o \left(\d_i / d_o\right)
Quick Summary
The Magnification Calculator evaluates linear optical magnification ($M = -\d_i / d_o = \h_i / h_o$) and resulting image height ($h_i$) from object height ($h_o$), object distance ($d_o$), and image distance ($d_i$).
Formula Explanation
Linear Magnification: M = -\d_i / d_o
Image Height: h_i = M \cdot h_o = -h_o \left(\d_i / d_o\right)
How It Works
The Magnification Calculator divides negative image distance ($-d_i$) by object distance ($d_o$) to determine linear magnification ratio ($M$). It multiplies $M$ by object height ($h_o$) to output image height ($h_i$) in cm, mm, absolute scale percentage ($|M| \times 100\%$), and orientation (inverted vs upright).
Step-by-Step Worked Example
Practical Problem: An object of height $h_o = 5.0\text{ cm}$ is placed $d_o = 30.0\text{ cm}$ from a lens, producing a real image focused at $d_i = +15.0\text{ cm}$. Calculate magnification $M$ and image height $h_i$.
- Step 1: Identify Input Parameters: $h_o = 5.0\text{ cm}$, $d_o = 30.0\text{ cm}$, $d_i = +15.0\text{ cm}$.
- Step 2: Apply the Linear Magnification Formula ($M = -d_i / d_o$): $M = -\15.0 / 30.0 = -0.5000\text{ (0.50x scale factor)}$.
- Step 3: Calculate Image Height ($h_i = M \cdot h_o$): $h_i = -0.5000 \times 5.0\text{ cm} = -2.500\text{ cm}$.
- Step 4: Determine Image Orientation & Nature: Since $M = -0.50 < 0$ (and $h_i = -2.5\text{ cm} < 0$), the image is **INVERTED** (upside down); since $d_i = +15.0\text{ cm} > 0$, the image is **REAL**.
- Step 5: Format Final Magnification Metrics: Magnification $M = -0.500x$. Image Height $h_i = -2.50\text{ cm}$ (-25.0 mm). Absolute Size = $50.0\%$ of original object height.
Real-World Calculation Examples
Scenario 1: Camera Sensor Image Reduction
Parameters: $h_o = 5\text{ cm}$, $d_o = 30\text{ cm}$, $d_i = +15\text{ cm}$
Result: $M = -0.50x$, $h_i = -2.50\text{ cm}$. Inverted half-size real image.
Scenario 2: Theater Cinema Projector 50x Enlargement
Parameters: $h_o = 3.5\text{ cm}$ (35mm film), $d_o = 10\text{ cm}$, $d_i = +500\text{ cm}$ (5m screen)
Result: $M = -50.00x$, $h_i = -175.00\text{ cm}$ (1.75m high screen image). Projector enlargement.
Scenario 3: Handheld Reading Magnifier 2.5x
Parameters: $h_o = 1\text{ cm}$ (text line), $d_o = 6\text{ cm}$, $d_i = -15\text{ cm}$ (virtual image)
Result: $M = +2.50x$, $h_i = +2.50\text{ cm}$. Upright enlarged virtual reading image.
Scenario 4: Security Convex Mirror Reduction
Parameters: $h_o = 180\text{ cm}$ (person), $d_o = 200\text{ cm}$, $d_i = -50\text{ cm}$
Result: $M = +0.25x$, $h_i = +45.00\text{ cm}$. Upright diminished virtual mirror image.
Key Benefits of Using This Calculator
Signed Height & Orientation Solver
Computes signed image height ($h_i$) and automatically reports inverted ($h_i < 0$) vs upright ($h_i > 0$) status.
Projector & Sensor Calibration
Essential for matching image sizes to camera sensors, film screens, and display panels.
Multi-Unit Readouts
Outputs magnification ratio ($M$), image height ($h_i$) in cm and mm, and percentage size scale.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is linear optical magnification?
Linear magnification (M) is the ratio of image height hi to object height ho, or negative image distance -di divided by object distance do (M = -di / do = hi / ho).
What does a negative magnification (-M) mean?
A negative sign (-) indicates an INVERTED (upside-down) image relative to the object.
What does a positive magnification (+M) mean?
A positive sign (+) indicates an ERECT (upright) image with the same vertical orientation as the object.
What does |M| > 1 vs |M| < 1 signify?
|M| > 1 means the image is ENLARGED (magnified); |M| < 1 means the image is DIMINISHED (reduced in size); |M| = 1 means equal size (1:1 scale).
What is angular magnification vs linear magnification?
Linear magnification M = hi / ho measures physical image height; angular magnification M_theta = theta_image / theta_object measures visual angle subtended at the eye (used in telescopes/microscopes).
What is total magnification in a compound microscope?
M_total = M_objective * M_eyepiece (e.g. 40x objective * 10x eyepiece = 400x total magnification).
How relates magnification to focal length f?
M = f / (f - do) = (f - di) / f.
Can a plane flat mirror magnify objects?
No! A flat mirror has di = -do, giving M = -(-do)/do = +1.0 (exact 1:1 scale, upright virtual image).
How converts magnification decimal to percentage scale?
Percentage Scale = |M| * 100% (e.g. M = -0.50 gives 50% scale size; M = +2.50 gives 250% scale size).
Why are real images always inverted for single thin lenses?
Because real images form on the opposite side of the lens (di > 0) through light ray crossover, resulting in M = -di / do < 0 (inverted).