Practice · Grade 9 Geometry
Lune of Hippocrates — Exercises
Work through lune-area problems using A = a·b/2 in rising difficulty, plus one boss question. Each task comes with a hint and a worked solution, free.
Q1 of 7
0 correct
A right triangle has legs a = 3 and b = 4. What is the combined area of the two lunes built on the legs?
A = a·b/2 = 3·4/2
Quick answer
What is the best way to practice the lune of Hippocrates?
Solve several problems with different legs a and b and always use the same formula: the two lunes on the legs together equal the triangle, so A = a·b/2. As a check, confirm the hypotenuse with c = √(a²+b²) each time, and remember that π disappears from the result. For a = 3, b = 4 the area is A = 6.
HowTo
A 3-step solving strategy
This order works for any problem given the two legs a and b of a right triangle.
- 1Step 1 of 3
Identify the legs and confirm the right angle
a and b are the two sides meeting at the right angle. To be safe, compute c = √(a²+b²) — for Pythagorean triples like 3-4-5 or 5-12-13 the value c is a whole number.
- 2Step 2 of 3
Form the triangle area: a·b/2
Multiply the two legs and divide by 2. That is the area of the right triangle — and by the theorem of Hippocrates also the combined area of the two lunes.
- 3Step 3 of 3
Apply the theorem of Hippocrates — no π
Write A(lune) = A△ = a·b/2. Do not try to compute the half-circles separately: when you subtract them, the Pythagorean theorem cancels every circular part.
Examples
Worked practice examples with full solutions
Four typical lune-area problems. Try each one yourself first, then compare with the solution.
Easy
Legs a = 3, b = 4. Find the lune area.
c = √(3²+4²) = √(9+16) = 5
A△ = a·b/2 = 3·4/2 = 6
A(lune) = A△ = 6
Check: c = 5 is a whole number → 3-4-5 triangle ✓
The classic example. The area follows directly from a·b/2 — no circles and no π.
Easy
Legs a = 5, b = 12. Find the lune area.
c = √(5²+12²) = √(25+144) = 13
A△ = a·b/2 = 5·12/2 = 30
A(lune) = 30
Check: 5-12-13 is a Pythagorean triple ✓
Here too c is a whole number. Compute a·b, then divide by 2 — that is enough.
Medium
Legs a = 6, b = 8. Find the lune area.
c = √(6²+8²) = √(36+64) = 10
A△ = a·b/2 = 6·8/2 = 24
A(lune) = 24
Check: 6-8-10 is the doubled 3-4-5 triple ✓
Scaling a 3-4-5 triangle by a factor of 2 quadruples the area: 6 → 24.
Hard
Boss: legs a = 9, b = 12. Find the lune area.
c = √(9²+12²) = √(81+144) = √225 = 15
A△ = a·b/2 = 9·12/2 = 108/2 = 54
A(lune) = A△ = 54
Check: 9-12-15 is the tripled 3-4-5 triple ✓
Three related triples (3-4-5, 6-8-10, 9-12-15). The formula a·b/2 stays the same throughout.
Pitfalls
Common mistakes — and how to avoid them
These five traps come up again and again when practicing the lune area.
Computing the half-circles separately
You do not need to go through the circle areas. The theorem of Hippocrates gives the area directly as a·b/2 — the circular parts cancel on their own.
Writing π into the result
Although half-circles are involved, π drops out of the final result entirely. An answer like "6π" is wrong; the correct answer is the plain number a·b/2.
Using the hypotenuse as a leg
a and b are the two short sides at the right angle. c = √(a²+b²) is the hypotenuse, not an input to the formula.
Counting only one lune
The theorem applies to the sum of both lunes on the legs. A single lune is not half the area of the triangle.
Forgetting to divide by 2
a·b is the area of a rectangle, not the triangle. The triangle area — and therefore the lune area — is a·b/2.
Study
Practice with a plan — three short tips
Know the Pythagorean triples by heart
3-4-5, 5-12-13, 8-15-17 and their multiples (6-8-10, 9-12-15) show up constantly. Knowing them lets you check the hypotenuse in your head and save time.
Solve first, then look at the answer
Write down your working before you reveal the hint. Active recall is far more effective for learning than passive reading.
For every wrong answer, ask why
Was it a forgotten ÷2? A stray π? A confused hypotenuse? Note the cause — next time you will spot the mistake at once.
FAQ
Frequently asked practice questions
Glossary
Terms in one sentence
- Lune
- A crescent-shaped area between two circular arcs — here between a half-circle on a leg and the half-circle on the hypotenuse.
- Leg
- One of the two sides at the right angle; a and b are the input values.
- Hypotenuse
- The longest side, opposite the right angle: c = √(a²+b²).
- Theorem of Hippocrates
- The two lunes on the legs together equal the triangle: A = a·b/2.
- Pythagorean theorem
- The rule a² + b² = c²; it is what makes all circular parts cancel in the lune proof.
- Half-circle
- Half of a circle's area, drawn outward over a side of the triangle.
- Quadrature
- Representing a curved-boundary area exactly by a straight-edged one.