Area & Arc Length
in Polar Coordinates
An interactive guide to understanding, deriving, and computing polar area and arc length — with live visualizations.
Instead of describing a point by its horizontal and vertical distances \((x,y)\), polar coordinates describe it by a distance from the origin \(r\) and an angle from the positive \(x\)-axis \(\theta\).
The conversion between the two systems is:
A polar curve is expressed as \(r = f(\theta)\): the radius changes as the angle sweeps around.
Try it: Drag the sliders to move the point around the polar plane. Notice how \(r\) sets the distance and \(\theta\) sets the direction.
In Cartesian coordinates we slice regions into thin rectangles. In polar coordinates the natural slice is a thin sector (a pie-slice shape), because the coordinates sweep angularly.
A sector with a tiny angular width \(d\theta\) and radius \(r\) has area:
$$dA = \frac{1}{2}r^2\,d\theta$$This is just the formula for a circle sector \(A=\tfrac{1}{2}r^2\theta\) with \(\theta\to d\theta\). Adding up all infinitesimal sectors from \(\alpha\) to \(\beta\) gives the total area.
The interactive below shows this Riemann-sum approximation. Increase \(n\) to watch the sectors converge to the true area.
- For a full curve that closes on itself, integrate over the full period (e.g.\ \(0\) to \(2\pi\) for a cardioid).
- For a petal of a rose, find the angles where \(r=0\) and integrate between them.
- For the area between two curves, use \(A=\frac{1}{2}\int_\alpha^\beta(r_{\text{outer}}^2 - r_{\text{inner}}^2)\,d\theta\).
Expand full derivation
Set up a Riemann sum. Divide \([\alpha,\beta]\) into \(n\) equal sub-intervals, each of angular width \(\Delta\theta = \tfrac{\beta-\alpha}{n}\). Choose a sample angle \(\theta_i^*\) in each sub-interval.
Area of one sector. The \(i\)-th sector is nearly a circular sector of radius \(r_i^* = r(\theta_i^*)\) and angle \(\Delta\theta\). A sector of a full circle of radius \(R\) and angle \(\phi\) has area \(\frac{1}{2}R^2\phi\), so:
$$\Delta A_i \approx \frac{1}{2}\bigl[r(\theta_i^*)\bigr]^2\,\Delta\theta$$Sum all sectors.
$$A \approx \sum_{i=1}^{n} \frac{1}{2}\bigl[r(\theta_i^*)\bigr]^2\,\Delta\theta$$Take the limit \(n\to\infty\). As \(\Delta\theta\to 0\) the sum becomes a definite integral by the definition of the Riemann integral:
$$A = \lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{2}\bigl[r(\theta_i^*)\bigr]^2\Delta\theta = \frac{1}{2}\int_{\alpha}^{\beta}\bigl[r(\theta)\bigr]^2\,d\theta \qquad \blacksquare$$Select a curve and drag the angle sliders to define the integration region. The shaded area and its numerical value update in real time.
Arc length is computed by summing infinitesimal chord lengths along the curve. In polar coordinates the position vector \(\mathbf{r}\) traces a path as \(\theta\) changes. Converting to parametric form:
$$x(\theta) = r(\theta)\cos\theta, \qquad y(\theta) = r(\theta)\sin\theta$$The arc length formula for parametric curves gives us exactly the polar arc length formula after simplification (see derivation below).
- Both \(r^2\) and \((r')^2\) must be non-negative (they always are), so the integrand is always real.
- For a circle \(r=a\), we get \(r'=0\) and \(L = \int_0^{2\pi}a\,d\theta = 2\pi a\) ✓
- This integral often lacks a closed form; numerical methods (like Simpson's rule) are frequently needed.
Expand full derivation
Parametric arc length. For any parametric curve \((x(\theta),y(\theta))\), the arc length from \(\alpha\) to \(\beta\) is:
$$L = \int_{\alpha}^{\beta}\sqrt{\left(\frac{dx}{d\theta}\right)^{\!2}+\left(\frac{dy}{d\theta}\right)^{\!2}}\;d\theta$$Differentiate the conversion formulas. Using \(x=r\cos\theta\) and \(y=r\sin\theta\) with the product rule:
$$\frac{dx}{d\theta} = r'\cos\theta - r\sin\theta, \qquad \frac{dy}{d\theta} = r'\sin\theta + r\cos\theta$$Square and add.
$$\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2$$ $$= (r')^2\cos^2\theta - 2rr'\cos\theta\sin\theta + r^2\sin^2\theta$$ $$\quad + (r')^2\sin^2\theta + 2rr'\sin\theta\cos\theta + r^2\cos^2\theta$$The cross terms cancel. Group the remaining terms:
$$= (r')^2(\cos^2\theta+\sin^2\theta) + r^2(\sin^2\theta+\cos^2\theta) = (r')^2 + r^2$$Substitute back.
$$L = \int_{\alpha}^{\beta}\sqrt{r^2+(r')^2}\;d\theta = \int_{\alpha}^{\beta}\sqrt{r(\theta)^2+\left(\frac{dr}{d\theta}\right)^{\!2}}\;d\theta \qquad \blacksquare$$The highlighted segment shows the arc between angles \(\alpha\) and \(\beta\). The computed length is shown below.
Shape: thin circular sector (pie-slice) of radius \(r\) and angle \(d\theta\)
Shape: hypotenuse of a right triangle with legs \(r\,d\theta\) and \(dr\)
Area inside the cardioid \(r = 1+\cos\theta\)
The cardioid completes one full revolution for \(\theta\in[0,2\pi]\).
$$A = \frac{1}{2}\int_0^{2\pi}(1+\cos\theta)^2\,d\theta$$Use \(\cos^2\theta = \tfrac{1+\cos 2\theta}{2}\):
$$= \frac{3}{2} + 2\cos\theta + \frac{\cos 2\theta}{2}$$Area of one petal of \(r = \cos(2\theta)\)
One petal lies where \(r\ge 0\) in \([-\pi/4,\pi/4]\) (the rightward petal).
$$A = \frac{1}{2}\int_{-\pi/4}^{\pi/4}\cos^2(2\theta)\,d\theta$$Area between two curves
Find intersection: \(2 = 1+\cos\theta \Rightarrow \theta = \pm\pi/2\).
$$A = \frac{1}{2}\int_{-\pi/2}^{\pi/2}\bigl(4-(1+\cos\theta)^2\bigr)\,d\theta$$Expanding and integrating:
$$= \frac{1}{2}\int_{-\pi/2}^{\pi/2}\!\!\!\left(\frac{5}{2}-2\cos\theta-\frac{\cos 2\theta}{2}\right)d\theta = \boxed{\frac{5\pi}{4}-2 \approx 1.927}$$Arc length of the circle \(r = 3\)
\(r = 3\), \(r' = 0\), so:
$$L = \int_0^{2\pi}\sqrt{9+0}\,d\theta = 3\cdot 2\pi = \boxed{6\pi}$$As expected: circumference of a circle of radius 3.
Arc length of the cardioid \(r = 1+\cos\theta\)
Use \(2+2\cos\theta = 4\cos^2(\theta/2)\):
$$\sqrt{r^2+(r')^2} = 2\left|\cos\tfrac{\theta}{2}\right|$$Arc length of the Archimedean spiral \(r = \theta\) on \([0,2\pi]\)
\(r=\theta\), \(r'=1\):
$$L = \int_0^{2\pi}\sqrt{\theta^2+1}\,d\theta$$Use the formula \(\int\sqrt{u^2+1}\,du = \tfrac{1}{2}(u\sqrt{u^2+1}+\sinh^{-1}u)+C\):
$$L = \frac{1}{2}\Big[\theta\sqrt{\theta^2+1}+\ln(\theta+\sqrt{\theta^2+1})\Big]_0^{2\pi}$$ $$\approx \boxed{21.256}$$Use the clickable keyboard to define curves — no typing needed. Add multiple curves to the same graph, reorder them, then compute the area between any two.