---
title: "Radius of curvature of the astroid"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:curvature-astroid-any-point"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-astroid-any-point
site: BCT Engineering
---

# Radius of curvature of the astroid

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2069 Chaitra
- Video: [radius of curvature of asteroid in parametric form.](https://www.youtube.com/watch?v=Ty5KM0bCulA) — Engineering Mathematics - Uma Raju
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-astroid-any-point
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Find the radius of curvature at any point (x,y) for the curve x^(2/3) + y^(2/3) = a^(2/3).

## Short answer

Using parametric or pedal form, ρ = 3a sin t cos t, which in cartesian coordinates is 3(a x y)^(1/3) (magnitude).

## Answer

Astroid: x^(2/3) + y^(2/3) = a^(2/3), with x = a cos³t, y = a sin³t.

Method 1 (parametric): 

```text
x′ = −3a cos²t sin t,   y′ = 3a sin²t cos t
x″ = 3a cos t (3 sin²t − 1),   y″ = 3a sin t (3 cos²t − 1)
x′² + y′² = 9a² sin²t cos²t
x′y″ − y′x″ = −9a² sin²t cos²t
```
|ρ| = (x′² + y′²)^(3/2) / |x′y″ − y′x″| = 27a³ sin³t cos³t / (9a² sin²t cos²t) = 3a sin t cos t.

Method 2 (pedal): from the pedal equation r² = a² − 3p², differentiate: 2r dr = −6p dp, so r dr/dp = −3p, hence ρ = r dr/dp = −3p, and |ρ| = 3p = 3a sin t cos t.

**In terms of x, y:** xy = a² sin³t cos³t, so a sin t cos t = (a x y)^(1/3).

**Answer: ρ = 3 (a x y)^(1/3) in magnitude (= 3a sin t cos t).**

## Check yourself

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

1. Which parametrisation is used for the astroid x^(2/3)+y^(2/3)=a^(2/3)?
   A. x = a cos^3 t, y = a sin^3 t
   B. x = a cos t, y = a sin t
   C. x = a cos^2 t, y = a sin^2 t
   D. x = a(t - sin t), y = a(1 - cos t)
   Answer: A. Since (cos^3 t)^(2/3) + (sin^3 t)^(2/3) = 1, this satisfies the curve equation. (Key point: Parametrisation)

2. The radius of curvature for a parametric curve is computed as
   A. (1 + y'^2)^(3/2) / y''
   B. (x'^2 + y'^2)^(3/2) / |x'y'' - y'x''|
   C. (x'^2 + y'^2)^(1/2) / |x'y'' - y'x''|
   D. (x'^2 + y'^2) / |x'y'' - y'x''|
   Answer: B. The standard parametric formula has the 3/2 power of x'^2 + y'^2 in the numerator. (Key point: Parametric formula)

3. The radius of curvature of the astroid is
   A. 3a sin^2 t cos^2 t
   B. a sin t cos t
   C. 3a sin t cos t
   D. 9a sin t cos t
   Answer: C. (27a^3 sin^3 t cos^3 t)/(9a^2 sin^2 t cos^2 t) = 3a sin t cos t. (Key point: Result in t)

4. Since xy = a^2 sin^3 t cos^3 t, the radius of curvature in x, y is
   A. 3(xy)^(1/3)
   B. 3a(xy)^(1/3)
   C. (axy)^(1/3)
   D. 3(axy)^(1/3)
   Answer: D. a sin t cos t = (a^3 sin^3 t cos^3 t)^(1/3) = (a x y)^(1/3), so rho = 3(axy)^(1/3). (Key point: In terms of x, y)

## Close variants set in other exams

- [Radius of curvature of a² = r²cos2θ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-lemniscate-a2-r2cos2theta) — Mathematics I, chapter 1, asked 2×

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