---
title: "Area of one loop of r = a sin 3θ"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:area-one-loop-sin-3theta"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/area-one-loop-sin-3theta
site: BCT Engineering
---

# Area of one loop of r = a sin 3θ

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2072 Kartik
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/area-one-loop-sin-3theta
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Find the area of one loop of the curve r = a sin 3θ.

## Short answer

One loop spans θ=0 to π/3. A = ½∫₀^{π/3} a² sin²3θ dθ = πa²/12 square units.

## Answer

Curve: r = a sin 3θ, a three-petal rose. r = 0 when 3θ = 0, π, 2π, ..., so θ = 0 and θ = π/3 bound one loop.

Area = ½ ∫ r² dθ over the loop:

```text
A = ½ ∫₀^(π/3) a² sin²3θ dθ
  = (a²/4) ∫₀^(π/3) (1 − cos 6θ) dθ
  = (a²/4) [θ − sin6θ/6]₀^(π/3)
  = (a²/4)(π/3 − 0)
```
**Area of one loop = πa²/12 square units.**

## Check yourself

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

1. The polar area formula is
   A. integral r dθ
   B. (1/2) integral r^2 dθ
   C. (1/2) integral r dθ
   D. π integral r^2 dθ
   Answer: B. Sector area. (Key point: Formula)

2. One loop of r = a sin 3θ lies between
   A. 0 and π
   B. 0 and π/6
   C. θ = 0 and π/3
   D. 0 and 2π/3
   Answer: C. r = 0 at 3θ = 0 and π. (Key point: Limits)

3. sin^2 3θ is replaced by
   A. (1 + cos 3θ)/2
   B. (1 - cos 3θ)/2
   C. 1 - cos 6θ
   D. (1 - cos 6θ)/2
   Answer: D. Half-angle identity. (Key point: Identity)

4. One loop area is
   A. πa^2/12
   B. πa^2/4
   C. πa^2/6
   D. πa^2/3
   Answer: A. (a^2/4)(π/3). (Key point: Result)

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