Goal. We form the lune as follows: draw a circle with centre O and radius R. Put points A and B on the circle so that triangle OAB is an isosceles right triangle with right angle at O (so OA = OB = R and ∠AOB = 90°). Draw the semicircle with diameter AB. The part of that semicircle lying outside the original circle is the lune. Show the area of the lune equals the area of triangle OAB.
Step 1 — compute some easy areas (write these down neatly in your working).
- Radius of the big circle: OA = OB = R, so the area of the big circle is πR².
- The central angle AOB is 90°, so the sector AOB (the slice of the big circle bounded by OA, OB and arc AB) is one quarter of the whole circle. Thus area(sector AOB) = (1/4)πR².
- Triangle OAB is right with legs OA = OB = R, so its area is (1/2)·R·R = R²/2.
- Length of AB (the hypotenuse) = R√2. The semicircle built on AB has radius r = (AB)/2 = (R√2)/2 = R/√2. So the area of that semicircle is (1/2)πr² = (1/2)π·(R²/2) = (πR²)/4.
Observation (very important): area of the semicircle on AB = area of the sector AOB = (πR²)/4. Keep that equality in mind.
Step 2 — identify the circular segment common to both figures.
Look at chord AB of the big circle. The sector AOB is made of two pieces: the triangle OAB plus the curved circular segment between arc AB and chord AB. In symbols:
area(sector AOB) = area(triangle OAB) + area(circular segment (arc AB, chord AB)).
Now look at the semicircle on AB. It also is made of two pieces: the part of that semicircle lying outside the big circle (that is the lune), plus exactly the same circular segment (the piece of the big circle bounded by arc AB and chord AB). So:
area(semicircle on AB) = area(lune) + area(circular segment (arc AB, chord AB)).
Why is the circular segment the same in both decompositions? Because the chord AB and the big-circle arc AB bound the same curved piece of the plane, and the semicircle (the circle having AB as diameter) passes through O as well (this follows from the fact that ∠AOB = 90° — so AB is a diameter of the circumcircle of triangle AOB). Therefore the overlap of the big circle and the semicircle that lies between arc AB and chord AB is exactly the circular segment mentioned above. (Draw a careful diagram and label points A, B, O and the midpoint M of AB to see the common region.)
Step 3 — subtract the common segment from the equal areas.
From Step 1 we know:
area(semicircle on AB) = area(sector AOB).
Using the decompositions from Step 2 we get:
area(lune) + area(circular segment) = area(triangle OAB) + area(circular segment).
Cancel the identical area(circular segment) from both sides, leaving:
area(lune) = area(triangle OAB).
Step 4 — check with numbers (optional but reassuring).
- area(triangle OAB) = R²/2.
- area(lune) = area(semicircle) − area(circular segment). But area(semicircle) = (πR²)/4 and area(circular segment) = area(sector) − area(triangle) = (πR²)/4 − R²/2. So
- area(lune) = (πR²)/4 − ((πR²)/4 − R²/2) = R²/2 = area(triangle).
Conclusion. The lune (the part of the semicircle outside the original circle) has the same area as triangle OAB. QED.
Teacher comments and rubric — Amy Chua "tiger mother" cadence (high expectations, direct):
- Student must hand in a clear diagram with O, A, B, midpoint M of AB, the big circle, and the semicircle. If your diagram is messy, do it again neatly — I will not accept sloppy diagrams.
- All steps must be written: compute areas of circle, sector, triangle, and semicircle. You must show the arithmetic (πR², πR²/4, R²/2, etc.).
- Explicitly state the two decompositions used: sector = triangle + segment and semicircle = lune + segment. Write one sentence explaining why the segment is the same in both decompositions (use the right-angle fact: ∠AOB = 90° implies the circle with diameter AB passes through O — Thales' theorem — so the semicircle meets the big circle at O, A, B and the bounded curved piece is identical).
- Rubric (marks out of 10):
- Diagram present and correct: 2 marks
- Correct computation of areas (circle, sector, triangle, semicircle): 3 marks
- Clear decomposition statements and identification of common segment: 2 marks
- Algebraic cancellation showing area(lune) = area(triangle): 2 marks
- Neatness, correct notation, and justification of Thales' theorem: 1 mark
- Teacher note (firm): I will deduct marks for missing labels, skipped justifications, or arithmetic left out. Redo until perfect.
ACARA v9 mapping (conceptual):
This task develops these Australian Curriculum (v9) mathematical competencies at a middle-secondary level (suitable for a 13-year-old working at the top of Year 7 / Year 8): geometric reasoning and proof, using circle properties, and area calculation. Key skills practiced:
- Use of circle properties (central angle, chord, Thales' theorem) to deduce geometric relationships.
- Decomposing shapes to compare areas (sector, triangle, semicircle, segment, lune).
- Application of algebraic cancellation and exact area formulas (πR², R²/2).
- Structured mathematical communication and justification (proof writing).
Additional enrichment / extensions (challenge ideas):
- Investigate other Hippocrates' lunes: find other angles or constructions where a lune has area equal to some polygonal region. Can you construct a lune equal to a given rectangle or triangle of different shape?
- Generalize: if the central angle ∠AOB = θ (not 90°), compare area of the sector to the area of the semicircle on AB — for which θ do you get interesting equalities? (Compute sector area = (θ/2π)πR² = (θ/2)R² and semicircle on AB has area (1/2)π·(AB/2)² where AB = 2R sin(θ/2); set up equality and explore.)
- Coordinate geometry approach: place O at (0,0), pick A and B, write equations of both circles, and compute the area of the lune using integrals — this strengthens calculus links (for older students).
- Constructive challenge: given a circle and an isosceles right triangle inside it (as above), can you construct with compass & straightedge another shape of equal area (for example, a rectangle)? Show the steps.
- Historical exploration: read about Hippocrates of Chios and the classical problem of squaring the circle; relate the lune construction to early attempts to understand areas bounded by circular arcs.
If you want, draw the diagram and send a photo of your working — I will mark it according to the rubric and tell you exactly what to fix. No sloppy work accepted; do it again until it is perfect.