Relation Between Focal Length (f) and Radius of Curvature (R)

Yashwant Parihar,

If you are studying Ray Optics in Class 10 or Class 12, you have probably come across the relation between focal length and radius of curvature and wondered why everyone keeps talking about the formula f = R/2. Don’t worry — you’re not alone. This is one of those concepts that looks confusing at first, but becomes really simple once you see how it works step by step.

In this post, I’ll walk you through the derivation of the relationship between the focal length and the radius of curvature of a spherical mirror in an easy, student-friendly way. We’ll use clear ray diagrams, simple explanations, and solved examples so you can understand the idea instead of just memorising the formula.

Whether you’re revising for your CBSE board exams or preparing for NEET or JEE, this guide will help you feel more confident about using the focal length formula and solving numerical problems related to mirrors and ray optics.

By the end, you’ll be able to write the derivation neatly in exams and apply f = R/2 without hesitation.

Relationship Between Focal Length and Radius of Curvature

For Concave Mirror:

f and R Relation in concave mirror

From △ MCN = Tan i = Perpendicular/ Base
Tan i = MN/CN Here, CN = -R (Distance Between Centre of Curvature and Pole)
Tan i = MN/-R
Tan i ~ i (For Small Angle)
i = MN/-R – (i)

From △ MFN = Tan 2i = Perpendicular/ Base
Tan 2i = MN/FN
Tan 2i = MN/-f Here, FN = -f (Distance between pole and focus)
Tan 2i ~ 2i (For Small Angle)
2i = MN/-f -(ii)

Putting the value of 1st in equation 2nd
2 x MN/-R = MN/-f
f = R/2 Hence Proved

For Convex Mirror:

f and R Relation in convex mirror

From △ MCN = Tan i = Perpendicular/ Base
Tan i = MN/CN Here, CN = R (Distance Between Centre of Curvature and Pole)
Tan i = MN/R
Tan i ~ i (For Small Angle)
i = MN/R – (i)

From △ MFN = Tan 2i = Perpendicular/ Base
Tan 2i = MN/FN
Tan 2i = MN/-f FN = f (Distance between pole and focus)
Tan 2i ~ 2i (For Small Angle)
2i = MN/f -(ii)
Putting the value of 1st in equation 2nd

2 x MN/R = MN/f
f = R/2 Hence Proved

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