LifeTally
Rectangle overlap and union area
What are the individual, shared and combined areas of your two axis-aligned rectangles?
Enter the lower and upper x and y bounds of each rectangle. The combined area counts every covered point once.
Your two rectangles
Use the same perpendicular Cartesian axes, equal scale and one coordinate unit for both rectangles. Their sides must be parallel to the axes. Enter decimal fractions, not a/b. All eight valid bounds are required for shared and combined areas.
How the estimate works
Each rectangle's area is its x span times its y span. Shared spans are max(0, smaller upper bound − larger lower bound) on each axis; multiply them for the shared area I. The combined area is U = A + B − I, so the shared region is counted once. Each individual area needs its own four valid bounds; shared area, combined area and overlap status need all eight, even beside a zero-area rectangle. Private exact decimal fractions of accepted numerical values are compared, subtracted, multiplied and summed before final numeric rounding. Rounded areas are never reused.
A = (Axmax − Axmin) × (Aymax − Aymin)
B = (Bxmax − Bxmin) × (Bymax − Bymin)
I = max(0, min(Axmax, Bxmax) − max(Axmin, Bxmin)) × max(0, min(Aymax, Bymax) − max(Aymin, Bymin))
U = A + B − IOnly two declared closed rectangles in one Euclidean plane, with sides parallel to the same coordinate axes. Equal bounds describe a valid zero-area limiting region. Zero shared area can mean separate regions, an edge or point contact, or a degenerate rectangle; it does not prove that the sets share no points. No rotated rectangles, arbitrary polygons, holes, three or more shapes, enclosing-box area, perimeter, overlap percentage, graph, 3D or geographic/survey area. No unit conversion, measurement validation, uncertainty estimate, material quantity or fit/safety guarantee. Exact relations between accepted numbers are not proof about measured objects.
OpenStax: rectangle area and square units ↗OpenStax: intersection, union and double counting ↗
CGAL: axis-parallel rectangle definition ↗
CGAL: intersection as common points ↗
Check the filled example
Our chosen example: A has x bounds −2 to 4 and y bounds 1 to 5, so its area is 6×4 = 24. B has x bounds 1 to 6 and y bounds −1 to 3, so its area is 5×4 = 20. Shared spans are 3 and 2: I = 6. The combined area is 24+20−6 = 38 coordinate units squared. These are mathematical example coordinates, not measurements or recommended dimensions.
Accepted input values
- Rectangle A: lower x bound
- -2
- Rectangle A: lower y bound
- 1
- Rectangle A: upper x bound
- 4
- Rectangle A: upper y bound
- 5
- Rectangle B: lower x bound
- 1
- Rectangle B: lower y bound
- -1
- Rectangle B: upper x bound
- 6
- Rectangle B: upper y bound
- 3
Plan the next step
Use the standard calculator for separate arithmetic. Check the coordinate order, axis alignment and consistent units before interpreting the areas.
Open the standard calculator →