A polar region is sliced into thin sectors rather than rectangles, because polar curves are organised around the origin.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The formula
The area is the integral of one half r² dθ. The one half comes from the area of a sector.
Do not forget the half
Omitting the one half doubles every answer. It is the most frequent slip in this topic.
Choosing the limits
The limits are angles, not x values. For one petal, integrate between the two angles where r is zero.
Between two curves
For the region between two polar curves, integrate one half of the outer r squared minus the inner r squared.
Watch for retracing
Integrating over a full turn on a rose with even n counts each petal once, but on odd n it retraces and doubles the answer.
Sectors instead of rectangles
Polar area is built from thin circular sectors rather than rectangles. A sector of angle dθ and radius r has area ½r²dθ, and integrating that gives the polar area formula.
The formula and its half
Area = ½∫r²dθ. The factor of a half and the squaring both come from the sector area, not from anywhere else. Omitting the half is the single most common error in polar area problems.
The limits trace the region once
Choose θ values that sweep the region exactly once. Sweeping twice doubles the answer, which happens easily with roses and cardioids. Sketching before integrating is what sets the limits correctly.
Area between polar curves
Integrate ½(R² − r²) where R is the outer curve. As with Cartesian regions, the subtraction happens inside the integral, and the intersection points give the limits.
Step 2: Try It Yourself
Tap and try it out.
- Diameter8 cm
- Circumference25.13 cm
- Circumference ÷ diameter3.14
Change the size. The circle gets bigger, but circumference divided by diameter stays at about 3.14 every time. That number is π.
Step 3: Watch an Example
One step at a time.
Watch Sana Set Up a Polar Area
Sana finds the area enclosed by the circle r = 4.
- Step 1
The formula is the integral of one half r² dθ.
Step 4: Your Turn
Practice makes it stick.
The Integrand
Problem 1 of 2
For r = 4, what is one half r squared?
The Area
Problem 2 of 2
The circle r = 4 encloses an area of kπ. What is k?
Sum the Sectors
1 of 8
For r = 6, what is one half r squared?
2 of 8
The circle r = 3 encloses kπ. What is k?
3 of 8
What fraction sits in front of r squared in the polar area formula?
4 of 8
The limits of a polar area integral are what? 1 angles, 2 x values.
5 of 8
The circle r = 5 encloses kπ. What is k?
6 of 8
For an area between curves, you subtract what? 1 the r values, 2 the r squared values.
7 of 8
Put the polar area setup in order.
- 1Find the angles where the region begins and ends.
- 2Write the integral of one half r squared d theta.
- 3Evaluate between those angles.
- 4Identify the region and the curve bounding it.
8 of 8
For r = 2, what is one half r squared?
Step 5: Quick Check
Show what you know.
Question 1 of 2
The circle r = 7 encloses an area of kπ. What is k?
Question 2 of 2
Where does the one half in the polar area formula come from?
What You Learned
- Polar area integrates one half r² dθ.
- The limits are angles, found where the region begins and ends.
- For a region between curves, subtract the squared radii.