The Area of an Orthodiagonal quadrilateral

An orthodiagonal quadrilateral is a quadrilateral with perpendicular diagonals (Josefsson, 2012). It has also been called a 'perpendicular' quadrilateral in the hierarchical classification in my (1994, 2009) book Some Adventures in Euclidean Geometry, which is available as a free download at the preceding link.

Problem
How can we find the area of an orthodiagonal quadrilateral?

Since rhombi and kites are obviously special cases of orthodiagonal quadrilaterals, it should not come as much of a surprise that the area of an orthodiagonal quadrilateral is the same, namely, (ABCD) = ½ AC x BD as shown in the dynamic sketch below.

Exploration
Explore the sketch by dragging to convince yourself that the formula is correct.
Drag ABCD into the shape of a concave quadrilateral. Is the formula still correct?
Drag ABCD into the shape of a crossed quadrilateral. Is the formula still correct?

The area of an orthodiagonal quadrilateral

Challenge
Can you explain why (prove that) the formula given above is correct?

Hint: Consider how you would prove the same formula for rhombi and/or kites and simply generalize the proof. Ensure that your proof is also valid for the concave case. The only tricky part is proving the result for the crossed case. But this might help you: Read the paper Finding the Area of a Crossed Quadrilateral.
Check: Compare your proof with the one given in De Villiers (1994, 2009) in the references below.
Alternative Proof: Show that the result is also a special case of the following more general formula: Area Formula for Quadrilateral in terms of its Diagonals.

Exploring Special Cases
Explore on your own, preferably using dynamic geometry, whether some the following quadrilaterals can also have perpendicular diagonals: isosceles trapezia, cyclic quadrilaterals, trapezia, tangential quadrilaterals, equidiagonal quadrilaterals, etc.

Another property
Another important property of an orthodiagonal quadrilateral is the following:
ABCD is an orthodiagonal quadrilateral, if and only if, the sums of the squares of the two pairs of opposite sides are equal, i.e. AB² + CD² = BC² + AD² (De Villiers, 1994; Maynard & Leversha, 2004; Josefsson, 2012).
You can view a dynamic sketch illustrating this result at: Pythagorean & Orthodiagonal Quadrilaterals & Some Properties (click on 'Link to Orthodiagonal quad')

Two More Challenges
a) Prove that a tangential (circumscribed) quadrilateral with perpendicular diagonals is a kite.
(Compare your own solution to Solution a).
b) Prove that a cyclic quadrilateral with equal diagonals is an isosceles trapezium (trapezoid).
(Compare your own solution to Solution b).

Some Pedagogical Comments
Pedagogically a teacher can create an opportunity to engage students with the so-called discovery function of proof by first letting students derive an area formula for a rhombus and/or a kite. By then asking them in the style of Polya to reflect on their proofs, and specifically whether their proofs would still apply if the quadrilateral had no symmetry, but the diagonals remained perpendicular, it is likely that some students would 'see' that the formula immediately generalizes to any orthodiagonal quadrilateral.
.....However, unlike a rhombus (which is only convex) or a kite (which can be convex or concave), an ortodiagonal quadrilateral can also become crossed. So in order to explain why the formula still works for the crossed case naturally leads to an excellent opportunity to introduce students to the concept of 'directed areas'.

References
De Villiers, M. (1994; 2009). Solution 12, Chapter 2, area orthodiagonal quad, pp. 122-124; Solution 4, Chapter 4, sum squares orthodiagonal quad, p. 153, Some Adventures in Euclidean Geometry (free to download). Lulu Publishers: Dynamic Mathematics Learning.
De Villiers, M. (2014). Finding the Area of a Crossed Quadrilateral. Scottish Mathematical Council Journal, 44 (Dec), pp. 71-74.
Josefsson, M. (2012). Characterizations of orthodiagonal quadrilaterals. Forum Geom., 12, pp. 13-25.
Maynard, P. & Leversha, G. (2004). Pythagoras' theorem for quadrilaterals. The Mathematical Gazette, Vol. 88, (March), pp. 128-130.

External Links
Orthodiagonal quadrilateral (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 17 March 2026.