Polar curves, area and arc length

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Some curves are far simpler in polar form, where a point is given by its distance r from the pole and the angle θ of its direction. A cardioid, a spiral or a rose is one short equation in r and θ. This lesson sketches polar curves from their equations, finds the area swept by the radius using thin sectors, avoids counting a region twice when a curve is traced more than once, and finds arc length in polar form.

A point from a distance and an angle

In polar coordinates a point is given by its distance r from the pole O and the angle θ that OP makes with the initial line, measured anticlockwise. The Cartesian coordinates are x = r cos θ, y = r sin θ, so r² = x² + y².

A polar curve r = f(θ) is traced as θ runs through a stated interval, here 0 ≤ θ < 2π, -π < θ ≤ π or part of one, with r ≥ 0 throughout. To sketch one:

tabulate r at the quarter turns and any angle where r = 0 or r is largest;
use symmetry: if f(-θ) = f(θ) the curve is symmetric about the initial line;
at a θ where r = 0, the curve reaches the pole in the direction θ.

For the cardioid r = 1 + cos θ, 0 ≤ θ < 2π: r = 2 at θ = 0, r = 1 at θ = π/2 and 3π/2, and r = 0 at θ = π. Since cos(-θ) = cos θ, it is symmetric about the initial line, and r ≥ 0 everywhere.

The cardioid r = 1 + cos θ

A heart-shaped curve with its cusp at the pole.

As θ increases from 0 to π, r falls from 2 to 0 and the upper half is traced; from π to 2π the lower half is traced by symmetry.

Figure data

This table is included in print while it is open.

Values of r = 1 + cos θ
θ	r
0	2
π/2	1
2π/3	1/2
π	0
3π/2	1

Area sweeps out thin sectors

A thin sector with angle δθ and radius r has area about (1/2)r² δθ. Adding these and letting δθ → 0 gives the area swept out by the radius as θ goes from α to β:

A = 1/2∫α^β r² dθ.

The formula measures area swept by the moving radius, so each part of the region must be swept exactly once.

Find the area enclosed by r = a(1 + cos θ), 0 ≤ θ < 2π, where a > 0.

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Square r and rewrite cos²θ
Method
Expand r² and use cos² θ = (1/2)(1 + cos 2θ).
Reason
Each resulting term has a standard integral.
Working
Answer
r² = a²(1 + 2 cos θ + cos² θ) = a²(3/2 + 2 cos θ + 1/2 cos 2θ).
Integrate once round the curve
Method
Integrate from 0 to 2π, which traces the curve exactly once.
Reason
Over a full turn, cos θ and cos 2θ integrate to zero.
Working
Answer
A = 1/2∫₀^2πa²(3/2 + 2 cos θ + 1/2 cos 2θ)dθ = (1/2)a² · 3/2 · 2π = (3π a²)/2.

Quick check

Using symmetry, 2 × 1/2∫₀^π r² dθ gives the same result. The cardioid fits inside the circle r = 2a of area 4π a², and (3/2)π a² is less than that, as it must be.

A student finds the area enclosed by r = cos θ as 1/2∫₀^2π cos² θ dθ = π/2. The curve is a circle of diameter 1. Explain the error and give the correct area.

Write your answer, then compare it with the solution

The error and the correct area

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Find where r is not negative
Method
cos θ ≥ 0 only for -π/2 ≤ θ ≤ π/2.
Reason
This syllabus uses curves with r ≥ 0; over a full turn, cos² θ sweeps the same circle a second time.
Working
Answer
A = 1/2∫_(-π/2)^(π/2) cos² θ dθ = 1/2 · π/2 = π/4.

Common mistake

A circle of diameter 1 has area π(1/2)² = π/4, which the student’s π/2 doubles. Before integrating, state the interval on which r ≥ 0 and the curve is traced once.

Find the exact area swept out by the radius of the spiral r = θ as θ increases from 0 to π.

Area, 3 significant figures

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A = 1/2∫₀^πθ² dθ = 1/2 · π³/3 = π³/6 ≈ 5.17.

The spiral never retraces itself on this interval, because r increases with θ, so no part of the region is counted twice.

Arc length in polar form

With x = r cos θ and y = r sin θ, the arc-length formula for a parametric curve simplifies, because (dx/(dθ))² + (dy/(dθ))² = r² + (dr/(dθ))²:

s = ∫α^β square root of (r² + (dr/(dθ))²) dθ.

Common mistakes

Using ∫ r dθ or forgetting the factor 1/2 in A = 1/2∫ r² dθ.
Integrating over an interval in which the curve is traced twice, so that the area is doubled.
Ignoring values of θ where r would be negative, which this syllabus excludes from a curve’s domain.
Writing the polar arc length as ∫ square root of (1 + (dr/(dθ))²) dθ; it is ∫ square root of (r² + (dr/(dθ))²) dθ.
Using degrees instead of radians for θ in an integral.

Before you move on

Show that the total length of the cardioid r = a(1 + cos θ), 0 ≤ θ < 2π, is 8a.

Write your answer, then compare it with the solution

Your simplification and integral

Hint 1

Use a half-angle identity

1 + cos θ = 2 cos² (θ/2). After taking the square root, check the sign of cos(θ/2) on your interval.

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Simplify the integrand
Method
dr/dθ = -a sin θ.
Reason
A perfect square under the root makes the integral elementary.
Working
Answer
r² + (dr/(dθ))² = a²(1 + 2 cos θ + cos² θ + sin² θ) = 2a²(1 + cos θ) = 4a² cos² θ/2.
Keep the absolute value and use symmetry
Method
square root of (4a² cos² (θ/2)) = 2a|cos(θ/2)|.
Reason
cos(θ/2) is negative for π < θ < 2π, so integrating 2a cos(θ/2) from 0 to 2π would give 0.
Working
Answer
Use the symmetric upper half:
s = 2∫₀^π2a cos θ/2 dθ = 2[4a sin θ/2]₀^π = 8a.

Quick check

A length of 0 is a clear warning that a square root has lost its sign. With a = 1, a calculator’s numerical integral of square root of (2 + 2 cos θ) from 0 to 2π also gives 8. Compare your reasoning with the solution; this answer is self-checked, not marked.

Syllabus and review details

GCE A-Level H2 Further Mathematics 2027 · 2027

Last reviewed: 2026-11-06

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Guided steps

Reading settings

A point from a distance and an angle

Area sweeps out thin sectors

The area inside a cardioid

Square r and rewrite cos²θ

Integrate once round the curve

A circle counted twice

Find where r is not negative

Area swept by a spiral

Arc length in polar form

Common mistakes

Before you move on

The perimeter of a cardioid

Simplify the integrand

Keep the absolute value and use symmetry

Lesson done

A point from a distance and an angle

Area sweeps out thin sectors

The area inside a cardioid

A circle counted twice

Area swept by a spiral

Arc length in polar form

Common mistakes

Before you move on

The perimeter of a cardioid

Questions