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How to find the area of the lunes of Hippocrates — step by step

Q: What are the lunes of Hippocrates?

Drawing a semicircle outward over each side of a right triangle creates crescent-shaped regions over the legs, sitting between the small semicircles and the large semicircle over the hypotenuse. These crescents are called lunes (Latin lunulae).

Q: Why do the two lunes equal the triangle?

Semicircle areas are proportional to the square of their diameter. When you add the leg semicircles and subtract the hypotenuse semicircle, the Pythagorean theorem (a²+b²=c²) makes all the circular parts cancel. What remains is a·b/2.

Q: Do I need π for the calculation?

No. Although semicircles are involved, π cancels out completely in the final result. The lune area is the purely algebraic quantity A = a·b/2.

Q: Does the triangle have to be right-angled?

Yes. Hippocrates' theorem assumes a right angle between the legs a and b, because only then does a²+b²=c² hold and the circular parts cancel.

Q: How do I find the hypotenuse?

By the Pythagorean theorem: c = √(a²+b²). For a = 3 and b = 4 this gives c = √(9+16) = 5. The hypotenuse is not an input — it follows from a and b.

Q: Is this related to squaring the circle?

Historically the lunes raised hopes: here a curved-edged region is represented exactly by a straight-edged one. Squaring the full circle, however, later turned out to be impossible.

Over a right triangle with legs a and b, draw a semicircle outward on each side. Hippocrates' theorem says the two crescent-shaped lunes over the legs together have exactly the area of the triangle, that is A = a·b/2. Worked example: a = 3, b = 4 → A = 6. Geometry for Grade 9 (advanced / enrichment).

By Valerij Hofmann · Fullstack Developer

Updated May 29, 2026Advanced

Quick answer

Draw a semicircle outward over each side of a right triangle, and the two lunes over the legs together have exactly the area of the triangle itself: A = a·b/2. For a = 3 and b = 4 this gives A = 6 — with no π in the answer.

At a glance

Summary of this tutorial
Example	a = 3, b = 4
Method	Hippocrates' theorem
Formula	A = a·b/2
Answer	A = 6
Check	c = √(9+16) = 5, A△ = 3·4/2 = 6 ✓
Grade level	Grade 9 (advanced / enrichment)

Worked example: a = 3, b = 4

EXAMPLE

a = 3, b = 4 → A = a·b/2

a and b are the legs at the right angle; we want the combined area of the two lunes.

The steps to the lune area

These steps work for any right triangle with legs a and b.

Step 1 · Start
a = 3, b = 4
The two legs at the right angle.
Step 2 · Pythagoras
c = √(a²+b²) = √(9+16) = 5
The large semicircle sits over the hypotenuse.
Step 3 · ·b/2
A△ = a·b/2 = 3·4/2
The area of the right triangle.
Step 4 · Hippocrates
A(lunes) = A△
The two lunes together equal the triangle.
Step 5 · Answer
= 6
The combined lune area — with no π.

Why the lunes equal the triangle

The area of a semicircle is proportional to the square of its diameter. Add the semicircles over the two legs and subtract the semicircle over the hypotenuse, and the Pythagorean theorem a² + b² = c² forces every circular part — and therefore π — to cancel exactly. What remains is the purely straight-sided triangle area a·b/2. So a region bounded by circular arcs is exactly squared by a straight-sided one.

Practice it yourself

Open the lune calculator

Enter your own legs a and b. See every step.

Start practice set

Lune-area questions

Frequently asked questions

What is the area of the lunes for a = 3 and b = 4?

It equals the triangle area: A = a·b/2 = 3·4/2 = 6. You don't need to work through the individual semicircles — Hippocrates' theorem gives the result directly.

What are the lunes of Hippocrates?

Why do the two lunes equal the triangle?

Do I need π for the calculation?

Does the triangle have to be right-angled?

How do I find the hypotenuse?

Is this related to squaring the circle?

Quick answer

At a glance

Worked example: a = 3, b = 4

The steps to the lune area

Step 1 · Start

Step 2 · Pythagoras

Step 3 · ·b/2

Step 4 · Hippocrates

Step 5 · Answer

Why the lunes equal the triangle

Practice it yourself

Frequently asked questions