---
title: "Area common to circle and cardioid"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:area-common-circle-cardioid"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/area-common-circle-cardioid
site: BCT Engineering
---

# Area common to circle and cardioid

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2074 Chaitra, 5 marks
- Video: [1.1.6 Area Double Integration Part 7 Engineering Maths 2 ,Find Area Outside Circle & Inside Cardioid](https://www.youtube.com/watch?v=AnUWGMCjPqQ) — Apna Tuition
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/area-common-circle-cardioid
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Find the area common to the circle r = a and the cardioid r = a(1+cosθ).

## Short answer

Intersection at θ=±π/2. Common area = a²(5π/4 − 2) = a²(5π−8)/4 square units.

## Answer

Curves: circle r = a and cardioid r = a(1 + cos θ). They meet when 1 + cos θ = 1, so cos θ = 0, θ = ±π/2.

By symmetry about the initial line, the common area is twice the area for 0 ≤ θ ≤ π.

- For 0 ≤ θ ≤ π/2: cardioid radius ≥ a, so the circle is the inner boundary (r = a).
- For π/2 ≤ θ ≤ π: cardioid radius ≤ a, so the cardioid is the inner boundary.

```text
Common area = 2 [ ½∫₀^(π/2) a² dθ + ½∫_(π/2)^π a²(1+cosθ)² dθ ]
            = a²·(π/2) + a² ∫_(π/2)^π (1 + 2cosθ + cos²θ) dθ
```
Evaluate: ∫_(π/2)^π 1 dθ = π/2; ∫_(π/2)^π 2 cos θ dθ = 2(0 − 1) = −2; ∫_(π/2)^π cos²θ dθ = π/4.

Sum = π/2 − 2 + π/4 = 3π/4 − 2.

Area = a² [π/2 + 3π/4 − 2] = a² (5π/4 − 2)

**Common area = a²(5π − 8)/4 square units.**

## Check yourself

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

1. The circle r=a and cardioid r=a(1+cos θ) meet at
   A. θ = 0
   B. θ = ±π/2
   C. θ = π
   D. θ = ±π/4
   Answer: B. 1 + cos θ = 1 gives cos θ = 0. (Key point: Intersection)

2. For 0 ≤ θ ≤ π/2 the inner boundary is
   A. the cardioid
   B. both
   C. the circle
   D. neither
   Answer: C. Cardioid radius ≥ a there. (Key point: Inner boundary)

3. For π/2 ≤ θ ≤ π the inner boundary is
   A. the circle
   B. both
   C. neither
   D. the cardioid
   Answer: D. Cardioid radius ≤ a there. (Key point: Inner boundary 2)

4. The common area is
   A. a^2(5π - 8)/4
   B. a^2(5π + 8)/4
   C. a^2(3π - 4)/4
   D. a^2(5π - 4)/8
   Answer: A. a^2(5π/4 - 2). (Key point: Result)

## Related in Mathematics I

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- [Volume of cardioid r=a(1+cosθ) about initial line](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-cardioid-1-plus-cos-initial-line) — Mathematics I, chapter 2, asked 6×
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- [Area under one arch of a cycloid](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aarea-cycloid-arc-and-base) — Mathematics I, chapter 2
- [Volume or surface area of cycloid about base](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-or-surface-cycloid-base) — Mathematics I, chapter 2

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