Polar Area Calculator

Area in polar coordinates comes from one integral, A = (1/2)∫ r² dθ, but applying it by hand means picking the right limits, squaring the function, and avoiding the double-counting traps that rose curves set. This polar area calculator does the work six ways: type any r(θ) and it integrates numerically between two angles, or skip the calculus and use the closed forms for circular sectors, annular sectors, cardioids, rose curves, and limaçons. Every result prints the formula it used, so you can check the arithmetic instead of trusting a black box.

Quick Answer

How to use the polar area calculator

Pick a calculation type, fill in its numbers, and press Calculate. Results stay hidden until you calculate; Reset restores the cardioid-style worked example in the general integrator.

1. General curve: type r(θ) with 1+cos(theta) pre-filled, adjust the radian limits (the degree boxes fill the radian ones for you), and press Calculate. The tool evaluates (1/2)∫ r² dθ with the midpoint rule over 2,000 subintervals, good to about six decimals for smooth curves.
2. Closed-form modes: choose sector, annulus, cardioid, rose, or limaçon and enter its parameters: sector needs radius and angle; annulus needs outer radius, inner radius (zero is allowed, giving a plain sector), and angle; cardioid needs a; rose needs amplitude a and a whole-number n; limaçon needs a and b.
3. Read the result line: the area prints at large size, sector and cardioid also report arc length (rθ and 8a), and the formula line shows the exact substitution, for example A = (3π/2)a² = 1.5 × π × 2² = 18.849556.
4. Watch the rejections: an end angle no larger than the start, a zero or negative radius, an inner radius reaching the outer radius, or an unparseable function each stops with a specific message instead of a wrong number.

Two honest limits of the function box: the parser accepts only theta (or θ), pi, sin, cos, tan, the four operators, ^ and parentheses. A string such as cat passes the character check but fails evaluation, and the tool says it could not parse the function rather than inventing a value. Decimal n in rose mode is truncated to an integer (2.9 becomes 2), so enter a whole number.

Worked example: cardioid r = 2(1 − cos θ)

Set the mode to Cardioid and type a = 2. The tool computes 3πa²/2 = (3π/2) × 4 = 6π = 18.849556 square units, and adds the arc length 8a = 16.000000. The formula line shows every substitution, so you can check it against the hand derivation in the next section.

The same numbers, produced by real runs of this tool rather than copied from a table:

For the cardioid check in reverse, switch to General curve with the pre-filled 1+cos(theta) from 0 to 2π and press Calculate: 4.712389, which is 3π/2, matching the closed form exactly. Type any row into the tool and it reproduces the figure.

Where the six formulas come from

All six modes are special cases of A = (1/2)∫ r² dθ. The closed forms are the integral evaluated analytically for each curve family, which is why they need no numerical error budget.

The cardioid form falls out of (1/2)∫₀^2π a²(1 − cos θ)² dθ = 3πa²/2, since the cos θ and cos² θ terms integrate to 0 and π respectively. The limaçon form is the same expansion with a and b: the cross term 2ab·cos θ vanishes over a full period and the cos² term contributes πb²/2, leaving π(a² + b²/2). That zero cross term is the detail some study pages hand-wave past, and it is exactly why the tool’s limaçon mode matches the integral it claims to be. The derivations follow the standard textbook treatment in Paul’s Online Math Notes on area with polar coordinates and the LibreTexts section on area and arc length in polar coordinates, which also establishes the cardioid arc length 8a the tool reports.

If you want the single-purpose version of two of these modes, the cardioid area calculator and the limacon area calculator each handle their own curve family, while this page is the six-mode workbench. The arc length calculator covers curve length for the general case, and the area between two functions calculator handles the rectangular-coordinate analogue of the between-curves problem.

The odd-n rose subtlety, and why tables get it wrong

For odd n, the 0–2π integral counts every petal twice, so the true enclosed area is πa²/4, half of what the naive integral says. Type r = 3·cos(3θ) into the general integrator from 0 to 2π and it returns about 14.137, which is 9π/2; the tool’s rose mode reports the correct enclosed area 7.068583 = 9π/4. Petal count alone never determines the area, which is where formula-sheet tables quietly fail: their entries of nπa²/4 for odd n and nπa²/2 for even n are only right when n = 1.

This is not a rounding quirk: for odd n the curve retraces each petal over the second half of the sweep, so the integral genuinely measures each region twice. The petal-count catalog in the LibreTexts polar graphs catalog confirms the n-petals-for-odd-n geometry behind the correction.

Using the general integrator well

The general mode evaluates (1/2)∫ r² dθ with the midpoint rule at 2,000 subintervals, which is accurate to about six decimal places for smooth curves. Sharp cusps and rapid oscillation (for example a high-frequency rose integrated over the wrong limits) can need narrower limits or a rethink of the setup.

Three practical notes. First, the degree boxes write into the radian boxes at six decimals, so 360 becomes 6.283185; the limits the formula line prints are the radian values the integral actually used. Second, the end angle must be strictly greater than the start: the tool refuses the degenerate zero-width integral with a message rather than returning 0. Third, if any of the 2,000 sample points cannot be evaluated (a function with a hole inside the interval, say), the result degrades to NaN rather than silently summing garbage; the visible failure is preferable to a plausible-looking wrong number.

What competing calculators miss

Polar area tools split into three camps, and each leaves something on the table:

• Single-formula tools that hide their convention: a well-known cardioid page publishes area = 6πa² with worked examples (a = 2 gives 75.36), without stating its polar equation. Under the r = a(1−cosθ) convention the correct closed form is 3πa²/2, giving 18.849556: check the formula a page assumes, because 6πa² is the correct area only for the r = 2a(1−cosθ) convention. The tool here derives the form from the integral and prints its substitution, so the convention cannot hide.
• Generic numerical integrators: a JavaScript-Math-expression integrator computes (1/2)∫ r² dθ with the trapezoidal rule but has no petal logic, no closed forms, and no degree convenience; it will happily double-count an odd-n rose without warning. This page pairs the integrator with the corrected closed forms so the two modes cross-check each other.
• Solvers and study tables: the big symbolic solvers integrate anything but gate steps behind accounts and are not polar-specific; the formula-sheet tables print nπa²/4 for odd-n roses, the exact double-count error documented above.
• Plotting sites: graphing tools draw the curves beautifully and compute no areas at all.
• The gap this page fills: six modes in one place, the general integrator whose pre-filled example reproduces the cardioid closed form to six decimals, the odd-n correction applied by default, arc length where a closed form exists, and every result showing the formula it used. Nothing here asks for an account or hides the arithmetic.

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