Pappus's area theorem

Key takeaways

Pappus's area theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pappus's area theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pappus's area theorem from memory before moving on to harder problems.

Reference excerpt

Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria (4th century AD), who discovered it.

Theorem Given an arbitrary triangle with two arbitrary parallelograms attached to two of its sides the theorem tells how to construct a parallelogram over the third side, such that the area of the third parallelogram equals the sum of the areas of the other two parallelograms. Let ABC be the arbitrary triangle and ABDE and ACFG the two arbitrary parallelograms attached to the triangle sides AB and AC. The extended parallelogram sides DE and FG intersect at H. The line segment AH now "becomes" the side of the third parallelogram BCML attached to the triangle side BC, i.e., one constructs line segments BL and CM over BC, such that BL and CM are a parallel and equal in length to AH. The following identity then holds for the areas (denoted by A) of the parallelograms:

A A B D E + A A C F G = A B C M L {\displaystyle {\text{A}}_{ABDE}+{\text{A}}_{ACFG}={\text{A}}_{BCML}}

The theorem generalizes the Pythagorean theorem twofold. Firstly it works for arbitrary triangles rather than only for right angled ones and secondly it uses parallelograms rather than squares. For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a square as well.

Proof Due to having the same base length and height the parallelograms ABDE and ABUH have the same area, the same argument applying to the parallelograms ACFG and ACVH, ABUH and BLQR, ACVH and RCMQ. This already yields the desired result, as we have:

A A B D E + A A C F G = A A B U H + A A C V H = A B L Q R + A R C M Q = A B C M L {\displaystyle {\begin{aligned}{\text{A}}_{ABDE}+{\text{A}}_{ACFG}&={\text{A}}_{ABUH}+{\text{A}}_{ACVH}\\&={\text{A}}_{BLQR}+{\text{A}}_{RCMQ}\\&={\text{A}}_{BCML}\end{aligned}}}

References Howard Eves: Pappus's Extension of the Pythagorean Theorem.The Mathematics Teacher, Vol. 51, No. 7 (November 1958), pp. 544–546 (JSTOR) Howard Eves: Great Moments in Mathematics (before 1650). Mathematical Association of America, 1983, ISBN 9780883853108, p. 37 (excerpt, p. 37, at Google Books) Eli Maor: The Pythagorean Theorem: A 4,000-year History. Princeton University Press, 2007, ISBN 9780691125268, pp. 58–59 (excerpt, p. 58, at Google Books) Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 77–78 (excerpt, p. 77, at Google Books)

External links The Pappus Area Theorem Pappus theorem

Illustrations

Worked examples

Example 1 — a first encounter with Pappus's area theorem

Start with the simplest possible case. Write down what Pappus's area theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pappus's area theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pappus's area theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pappus's area theorem

In research: Pappus's area theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pappus's area theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pappus's area theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ancient Greek mathematics, Area, Equations, so understanding it makes those chapters shorter.
In everyday life: Look for Pappus's area theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pappus's area theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pappus's area theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pappus's area theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pappus's area theorem in simple terms?

Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria (4th cen…

Why does Pappus's area theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pappus's area theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pappus's area theorem.