---
title: "Radius of curvature of r = a(1−cosθ)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:curvature-cardioid-1-minus-cos"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-cardioid-1-minus-cos
site: BCT Engineering
---

# Radius of curvature of r = a(1−cosθ)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2067 Asar
- Video: [Radius of Curvature | r=a(1+costheta)](https://www.youtube.com/watch?v=TR_YccZt6qM) — Learn With Sumi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-cardioid-1-minus-cos
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the radius of curvature of the curve r = a(1−cosθ).

## Short answer

By the pedal method, p² = r³/(2a); then ρ = r dr/dp = (2/3)√(2ar) = (4a/3) sin(θ/2).

## Answer

Cardioid: r = a(1 − cos θ) = 2a sin²(θ/2). Use the pedal method.

**Angle φ:** dr/dθ = a sin θ, so

tan φ = r dθ/dr = a(1 − cos θ)/(a sin θ) = tan(θ/2), so φ = θ/2.

**Pedal equation:** p = r sin φ = r sin(θ/2). From r = 2a sin²(θ/2), sin(θ/2) = √(r/2a). Hence

p = r · √(r/2a) = r^(3/2)/√(2a), that is p² = r³/(2a).

**Radius of curvature:** ρ = r dr/dp.

dp/dr = (3/2) r^(1/2)/√(2a)

ρ = r · √(2a) / ((3/2) r^(1/2)) = (2/3) √(2a) · √r

**Answer: ρ = (2/3)√(2ar) = (4a/3) sin(θ/2).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For r = a(1 - cos θ), tan(phi) = r dθ/dr equals
   A. tan(θ/2), so phi = θ/2
   B. tan θ, so phi = θ
   C. cot(θ/2), so phi = π/2 - θ/2
   D. -tan(θ/2), so phi = -θ/2
   Answer: A. r/(dr/dθ) = a(1 - cos θ)/(a sin θ) = tan(θ/2). (Key point: Angle phi)

2. The pedal equation of the cardioid is
   A. p^2 = r^2/(2a)
   B. p^2 = r^3/(2a)
   C. p = r^3/(2a)
   D. p^2 = 2a r^3
   Answer: B. p = r sin(θ/2) = r sqrt(r/2a), so p^2 = r^3/(2a). (Key point: Pedal equation)

3. In the pedal method the radius of curvature is
   A. rho = p dr/dp
   B. rho = r dp/dr
   C. rho = r dr/dp
   D. rho = dr/dp
   Answer: C. The pedal formula is rho = r dr/dp. (Key point: Pedal formula)

4. The radius of curvature is
   A. (3/2) sqrt(2ar)
   B. (2/3) sqrt(ar)
   C. 2 sqrt(2ar)
   D. (2/3) sqrt(2ar)
   Answer: D. dp/dr = (3/2) sqrt(r)/sqrt(2a), so rho = r/(dp/dr) = (2/3) sqrt(2ar). (Key point: Result)

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