Lens Math

Learn how to use equations with lens math to determine information about the image like height and distance from the lens.

Lens Math Variables

Name Variable Unit Unit Abbreviation
focal length f centimeters or meters cm or m
distance to image di centimeters or meters cm or m
distance to object do centimeters or meters cm or m
magnification M times x
height of image hi centimeters or meters cm or m
height of object ho centimeters or meters cm or m

Virtual and Real Sides of a Lens

Light travels through a lens unlike a mirror which reflects.  Light travels to the real side given a positive (+) which is on the opposite side of the object.   If the image appears on the real side it will be given a positive value.  If the image appears on the virtual side, side of the object, it is virtual and given a negative symbol.

Lens Math Facts

  • For all lenses:
    • do is always positive + no matter what, an object is always real
  • For a concave lens:
    • (only produce, virtual, upright, reduced images)
    • f is always negative (-)
      • concave lens focal distances will be negative during math since the curve of the front side is toward the virtual side.
    • di is always negative (-)
      • because concave lenses only produce virtual upright images
  • For a convex lens:
    • f is always positive (+)
    • convex lens focal distances will be positive since the curve of the front side is toward the real side.
    • di can be positive or negative.  You will determine this by the solution for di.
      • If di is a positive in your solution, the image is real and inverted.
      • If di is a negative in your solution, the image is virtual and upright.

Lens Equation Tips

Lens Equation

do

  • First of all the object is always real and do will always be positive, never make this negative and if a solution is negative, you did something wrong.

di

  • If a problem says you have a virtual image, make the di negative.  If a problem says that the image appears on the same side as the object, make the di negative.
  • A real image is always inverted, a virtual image is always upright.

f

  • If you have a concave lens the f will always be negative
  • If you have a convex lens the f will always be positive

Magnification Equation

This equation can be used three ways

  • (hi / h) = (di / -d)
  • M= hi / ho
  • M= -di / do

M

  • you can solve for M with hi and ho
    • M= hi / ho
  • you can solve for M with di and do
    • M= -di / do
  • If M is 1 the image is not magnified
  • If M is less than 1 the image is reduced
  • If M is greater than 1 the image is enlarged
  • If M is negative the image is real and inverted
  • If M is positive the image is virtual and upright

Example Problems

Use the video provided for extra support through these questions.

1. An object is 5.00 meters away from a convex lens, which produces a real image 1.00 meters away.  What is the focal length of the lens?

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2. An object is 5.00 meters away from a convex lens, which produces a real image 1.00 meters away.  What is the magnification of the image?

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3. An object is 6.0 centimeters away from a convex lens, which produces a virtual image 7.0 centimeters away.  What is the focal length of the lens?

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4. A 2 cm tall object is 6.0 centimeters away from a convex lens, which produces a virtual image 7.0 centimeters away.  What is the height of the image?

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5. An 8 cm tall object is 12.0 centimeters away from a concave lens, which produces a virtual image 7.0 centimeters away.  What is the focal length of the lens?

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6. An 8 cm tall object is 12.0 centimeters away from a concave lens, which produces a virtual image 7.0 centimeters away.  What is the height of the image?

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7. Find the image distance for a convex lens with an object distance of 15 cm and a focal length of 30 cm.

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8. Given di = -18 cm, do= 6 cm, and M=3, describe the image.

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9. Find the focal length for a convex lens with an object distance of 60 cm and an real image distance of 15 cm.

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10. Find the distance an object is away from a convex lens with an real image formed 60 cm away and a focal length of 20 cm.

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PhET Lens Lab

Return to the previous section by clicking here to complete a PhET lens lab that includes lens math.

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