Lexell's theorem

Key takeaways

Lexell's 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 Lexell's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lexell's theorem from memory before moving on to harder problems.

Reference excerpt

In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle, called Lexell's circle or Lexell's locus, passing through each of the two points antipodal to the two base vertices. A spherical triangle is a shape on a sphere consisting of three vertices (corner points) connected by three sides, each of which is part of a great circle, the analog on the sphere of a straight line in the plane (for example the equator and meridians of a globe); the spherical analog of planar circles, which curve relative to the surface, are called small circles (for example the circles of latitude other than the equator). Any of the sides of a spherical triangle can be considered the base, and the opposite vertex is the corresponding apex. Two points on a sphere are antipodal if they are diametrically opposite, as far apart as possible. The theorem is named for Anders Johan Lexell, who presented a paper about it c. 1777 (published 1784) including both a trigonometric proof and a geometric one. Lexell's colleague Leonhard Euler wrote another pair of proofs in 1778 (published 1797), and a variety of proofs have been written since by Adrien-Marie Legendre (1800), Jakob Steiner (1827), Carl Friedrich Gauss (1841), Paul Serret (1855), and Joseph-Émile Barbier (1864), among others. The theorem is the analog of propositions 37 and 39 in Book I of Euclid's Elements, which prove that every planar triangle with the same area on a fixed base has its apex on a straight line parallel to the base. An analogous theorem can also be proven for hyperbolic triangles, for which the apex lies on a hypercycle.

Statement

Given a fixed base A B , {\displaystyle AB,} an arc of a great circle on a sphere, and two apex points C {\displaystyle C} and X {\displaystyle X} on the same side of great circle A B , {\displaystyle AB,} Lexell's theorem holds that the surface area of the spherical triangle △ A B X {\displaystyle \triangle ABX} is equal to that of △ A B C {\displaystyle \triangle ABC} if and only if X {\displaystyle X} lies on the small-circle arc B ∗ C A ∗ , {\displaystyle B^{*~\!\!}CA^{*}\!,} where A ∗ {\displaystyle A^{*~\!\!}} and B ∗ {\displaystyle B^{*~\!\!}} are the points antipodal to A {\displaystyle A} and B , {\displaystyle B,} respectively. As one analog of the planar formula area = 1 2 base ⋅ height {\displaystyle {\text{area}}={\tfrac {1}{2}}\,{\text{base}}\cdot {\text{height}}} for the area of a triangle, the spherical excess ε {\displaystyle \varepsilon } of spherical triangle △ A B C {\displaystyle \triangle ABC} can be computed in terms of the base c {\displaystyle c} (the angular length of arc A B {\displaystyle AB} ) and "height" h c {\displaystyle h_{c}} (the angular distance between the parallel small circles A ∗ B ∗ C {\displaystyle A^{*~\!\!}B^{*~\!\!}C} and A B C ∗ {\displaystyle ABC^{*~\!\!}} ):

sin ⁡ 1 2 ε = tan ⁡ 1 2 c tan ⁡ 1 2 h c . {\displaystyle \sin {\tfrac {1}{2}}\varepsilon =\tan {\tfrac {1}{2}}c\,\tan {\tfrac {1}{2}}h_{c}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Lexell's theorem

Start with the simplest possible case. Write down what Lexell's 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 Lexell's 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 Lexell's 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 Lexell's theorem

In research: Lexell's 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 Lexell's 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: Lexell's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Area, Spherical trigonometry, Theorems about triangles and circles, so understanding it makes those chapters shorter.
In everyday life: Look for Lexell's 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 Lexell's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lexell's 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 Lexell's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lexell's theorem in simple terms?

In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle, called Lexell's circle or Lexell's locus, passing through each of the two points antipodal to the two base vertices. A spherical triangle is a shape…

Why does Lexell's 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 Lexell's 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 Lexell's theorem.