Calculate the area between two intersecting circles from their radii and center distance, or solve for a missing variable in the formula.

Area Between Two Intersecting Circles Calculator

Enter any 3 values to calculate the missing variable

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Area Between Two Intersecting Circles Formula

The area between two intersecting circles is the overlap area shared by both circles. This is also called the circle intersection area or lens area.

A = r1²*cos⁻1((d² + r1² - r2²) / (2*d*r1)) + r2²*cos⁻1((d² + r2² - r1²) / (2*d*r2)) - 0.5*sqrt((- d + r1 + r2)*(d + r1 - r2)*(d - r1 + r2)*(d + r1 + r2))
  • A = area of intersection between the two circles
  • r1 = radius of circle 1
  • r2 = radius of circle 2
  • d = distance between the centers of the two circles
  • cos-1 = inverse cosine, measured in radians

There are also two special cases:

If d > = r1 + r2, then A = 0
If d < = abs(r1 - r2), then A = pi*min(r1,r2)²
  • If the distance between centers is greater than or equal to the sum of the radii, the circles do not overlap, so the intersection area is 0.
  • If the distance between centers is less than or equal to the difference between the radii, the smaller circle is completely inside the larger circle, so the intersection area is the full area of the smaller circle.
  • For partial overlap, the calculator uses the full intersection formula and returns the shared area.
  • The calculator converts all length inputs to meters internally, calculates the area in square meters, then converts the result to your selected area unit.

Circle Overlap Cases

Condition Relationship Intersection Area
No overlap d >= r1 + r2 0
One circle inside the other d <= |r1 – r2| Area of the smaller circle
Partial overlap |r1 – r2| < d < r1 + r2 Lens-shaped shared area
Same center, same radius d = 0 and r1 = r2 Area of either circle

Common Length and Area Unit Conversions

Unit Length in meters Area in square meters
millimeter 1 mm = 0.001 m 1 sq mm = 0.000001 sq m
centimeter 1 cm = 0.01 m 1 sq cm = 0.0001 sq m
meter 1 m = 1 m 1 sq m = 1 sq m
inch 1 in = 0.0254 m 1 sq in = 0.00064516 sq m
foot 1 ft = 0.3048 m 1 sq ft = 0.092903 sq m

Example Problems

Example 1: Two equal circles with partial overlap

Find the area between two intersecting circles where:

  • r1 = 5 cm
  • r2 = 5 cm
  • d = 6 cm

Using the intersection formula:

A = 22.4071 sq cm

The shared area is approximately 22.4071 square centimeters.

Example 2: One circle completely inside another

Find the intersection area where:

  • r1 = 10 m
  • r2 = 4 m
  • d = 3 m

Since d <= |r1 – r2|, the smaller circle is completely inside the larger circle.

A = pi*4² = 50.2655 sq m

The intersection area is approximately 50.2655 square meters.

FAQs

What does the area between two intersecting circles mean?

It means the area shared by both circles. If the circles overlap partially, the shared region forms a lens shape. If one circle is completely inside the other, the shared area is the area of the smaller circle. If the circles do not touch or overlap, the shared area is 0.

Why does the distance between centers matter?

The distance between centers controls how much the circles overlap. A smaller distance usually means more overlap. A larger distance means less overlap. When the distance is greater than or equal to the sum of the radii, the circles no longer overlap.

Can the radii and distance use different units?

Yes. You can enter the radii and center distance using different supported length units. The calculator converts them to a common unit before calculating the intersection area, then converts the answer to the area unit you select.