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Rectangle Diagonal, Opposite-Long-Side Angle and Boundary

It is the right or obtuse angle opposite the longer side, 2asin(l/d). The other diagonal angle is supplementary. A square gives 90 degrees for both.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Longer side (l) 4, Shorter side (w) 3 gives Diagonal (d) 5, Area 12, Perimeter 14, Circumcircle radius (r) 2.5, Diagonal angle opposite longer side α 106.2602.

d² = l²+w²; A = lw; P = 2(l+w); α = 2asin(l/d)

Worked example

Enter these known values and leave the other values blank.

Longer side (l)
4 cm
Shorter side (w)
3 cm
Area
12 cm²
Perimeter
14 cm
Diagonal angle opposite longer side α
106.26 deg
Circumcircle radius (r)
2.5 cm
Diagonal (d)
5 cm

Assumptions and limitations

  • Positive longer side l is at least shorter side w. The angle reported opposite l between intersecting diagonals is right or obtuse, from 90° inclusive to 180° exclusive. Its hidden supplementary helper stays hidden. The reviewed domain includes squares. Perimeter/area selects the larger quadratic root for l. Two independent supported measures are needed; perimeter alone is insufficient.
  • Clear unknowns before solving backward and select compatible units. Original reference drawings do not scale to entered values. Independently rounded boundary measurements can be incompatible at finite precision.

Common questions

Which angle between the diagonals is displayed?

It is the right or obtuse angle opposite the longer side, 2asin(l/d). The other diagonal angle is supplementary. A square gives 90 degrees for both.

How many measurements are needed?

Use the worked example or a verified independent input pair or triple. Supplying dependent measurements does not add enough information; a consistent shape still needs a length scale.

Which input pairs support the retained inverse controls?

Use both sides; either side with area, perimeter or diagonal; the opposite-long-side diagonal angle with a side, perimeter or diagonal; perimeter with area or diagonal; or a circumcircle radius with one side. Area and diagonal alone do not populate the sides in this captured workflow.

References

Require positive physical rectangle measures and real inverse roots, with equality allowed for square rectangles. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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