Polar coordinates, explained for AP Calculus BC
Polar coordinates describe a point by its distance from the origin and its angle, instead of x and y. They make circles, spirals and petal shapes easy to write, and AP Calculus BC asks you to find slopes and areas for them.
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What polar coordinates are
A point in polar coordinates is written (r, θ): r is the distance from the origin (the pole), and θ is the angle measured anticlockwise from the positive x-axis, usually in radians. The point (2, π/3) is 2 units from the origin at an angle of 60°.
Unlike Cartesian coordinates, polar coordinates are not unique. (2, π/3) is the same point as (2, 7π/3), because adding a full turn changes nothing. A negative r means “go the opposite way”: (−2, π/3) is the same point as (2, 4π/3).
Converting between polar and Cartesian
- x = r cosθ and y = r sinθ
- r² = x² + y² and tanθ = y/x (check the quadrant)
Example 1: polar to Cartesian
Convert (4, 5π/6) to Cartesian coordinates.
- x = 4cos(5π/6) = 4(−√3/2) = −2√3. y = 4sin(5π/6) = 2.
- (−2√3, 2).
Example 2: Cartesian to polar
Convert (−1, −1) to polar form with r > 0.
- r = √2. tanθ = 1, but the point is in the third quadrant, so θ = 5π/4.
- (√2, 5π/4). A calculator would give π/4, which is the wrong quadrant.
Example 3: converting an equation
Write r = 4cosθ in Cartesian form.
- Multiply by r: r² = 4r cosθ, so x² + y² = 4x.
- Complete the square: (x − 2)² + y² = 4, a circle with centre (2, 0) and radius 2.
Common polar curves
- r = a: a circle of radius a centred at the origin.
- θ = α: a straight line through the origin.
- r = 2a cosθ or r = 2a sinθ: circles of radius a passing through the origin.
- r = a(1 + cosθ): a cardioid, a heart shape.
- r = a cos(nθ): a rose with n petals if n is odd, 2n petals if n is even.
- r = aθ: a spiral.
When sketching, make a quick table of r for θ = 0, π/4, π/2 and so on, and watch where r = 0, because those angles are where the curve passes through the origin. A graphing calculator in polar mode is a quick way to check your sketch.
Slopes of polar curves
A polar curve r = f(θ) is really a parametric curve with parameter θ: x = f(θ)cosθ and y = f(θ)sinθ. So
dy/dx = (dy/dθ) / (dx/dθ)
with dy/dθ = f'(θ)sinθ + f(θ)cosθ and dx/dθ = f'(θ)cosθ − f(θ)sinθ. Do not confuse dy/dx with dr/dθ: dr/dθ tells you whether the curve is moving towards or away from the origin, not its slope.
Example 4: a slope
Find dy/dx for r = 2 + 2cosθ at θ = π/2.
- f(π/2) = 2 and f'(θ) = −2sinθ, so f'(π/2) = −2.
- dy/dθ = (−2)(1) + (2)(0) = −2. dx/dθ = (−2)(0) − (2)(1) = −2.
- dy/dx = 1.
Example 5: interpreting dr/dθ
For r = 3 − 2sinθ, is the curve moving towards or away from the origin at θ = π/6?
- dr/dθ = −2cosθ, which is −√3 < 0 at π/6, and r = 2 > 0.
- r is positive and decreasing, so the curve is moving towards the origin.
Area enclosed by a polar curve
Area in polar coordinates is built from thin sectors, not rectangles. A sector of radius r and small angle dθ has area ½r² dθ, so
Area = ½ ∫αβ r² dθ
Example 6: area of a circle
Check the formula using r = 3 for 0 ≤ θ ≤ 2π.
- Area = ½ ∫02π 9 dθ = ½ × 9 × 2π = 9π, as expected for a circle of radius 3.
Example 7: a cardioid
Find the area enclosed by r = 1 + cosθ.
- Area = ½ ∫02π (1 + cosθ)² dθ = ½ ∫02π (1 + 2cosθ + cos²θ) dθ.
- Using cos²θ = ½(1 + cos 2θ), the integral is 2π + 0 + π = 3π.
- Area = 3π/2.
Area between two polar curves
For the region between an outer curve R(θ) and an inner curve r(θ), subtract: Area = ½ ∫αβ (R² − r²) dθ. Find the limits by setting the two curves equal, and sketch to see which is outer on each interval. As with washers, square each radius separately.
Example 8: between two curves
Find the area inside r = 3cosθ and outside r = 1 + cosθ.
- Intersections: 3cosθ = 1 + cosθ gives cosθ = 1/2, so θ = ±π/3. Between these angles the circle is outside the cardioid.
- Area = ½∫−π/3π/3 (9cos²θ − (1 + cosθ)²) dθ = ∫0π/3 (3 + 4cos 2θ − 2cosθ) dθ, using symmetry and cos²θ = ½(1 + cos 2θ).
- = [3θ + 2sin 2θ − 2sinθ]0π/3 = π + √3 − √3 = π.
Polar curves on the free-response section
AP BC free-response questions on polar curves usually combine several skills: find an area, use a calculator to evaluate an integral, find dy/dx or dr/dθ at a point, and interpret the sign of dr/dθ in words. When a calculator is allowed, write the integral expression first, with limits and the ½, then give the value to three decimal places. Points are often awarded separately for the setup and the answer, so a correct setup is valuable even if the arithmetic goes wrong. Practise with released questions from AP Central.
Common mistakes
- Using ∫ r dθ or forgetting the ½ in the area formula.
- Treating dr/dθ as the slope dy/dx.
- Getting the wrong quadrant when finding θ from tanθ.
- Using (R − r)² instead of R² − r² between two curves.
- Choosing limits that trace part of the curve twice.
Practice questions
Try each one before opening the answer.
Q1. Convert (2, π) to Cartesian.
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Q2. Convert x² + y² = 25 to polar form.
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Q3. How many petals does r = sin(3θ) have?
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Q4. Find the area enclosed by r = 2sinθ for 0 ≤ θ ≤ π.
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Frequently asked questions
Are polar coordinates on AP Calculus AB?
No. Polar curves, their slopes and areas are in Unit 9 of AP Calculus BC only.
Are polar coordinates in A level Maths?
Not in A level Maths. Polar coordinates appear in A level Further Maths, including areas enclosed by polar curves.
Key takeaways
- x = r cosθ, y = r sinθ, r² = x² + y².
- Polar coordinates are not unique; watch quadrants and negative r.
- dy/dx = (dy/dθ)/(dx/dθ), not dr/dθ.
- Area = ½∫ r² dθ.
- Between curves: ½∫ (R² − r²) dθ.