Goal: Show the lune (the part of the semicircle on the hypotenuse that lies outside the original circle) has the same area as the isosceles right triangle.
Setup and labels
Let O be the centre of the original circle. Place an isosceles right triangle OAB so that OA = OB = R and ∠AOB = 90°. Then AB is the triangle's hypotenuse. Draw the semicircle with diameter AB. The lune is the portion of this semicircle that lies outside the original circle centred at O.
Step 1 — Areas of semicircle on AB and the sector AOB
Length AB = R√2, so the radius of the semicircle on AB is (AB)/2 = R/√2. Its area is
Area(semicircle on AB) = (1/2)·π·(R/√2)^2 = (1/2)·π·(R^2/2) = πR^2/4.
The sector AOB in the original circle has central angle 90°, so
Area(sector AOB) = (90°/360°)·πR^2 = πR^2/4.
So the semicircle on AB and the sector AOB have exactly the same area.
Step 2 — Break each region into two parts
The sector AOB can be split into: (i) triangle OAB and (ii) the circular segment between arc AB (of the original circle) and chord AB. So
Area(sector AOB) = Area(triangle OAB) + Area(circular segment).
The semicircle on AB can be split into: (i) the lune (the part outside the original circle) and (ii) the same circular segment between arc AB and chord AB (this segment lies where the semicircle and the original circle overlap). So
Area(semicircle on AB) = Area(lune) + Area(circular segment).
(You can check by drawing chord AB and observing the same curved piece appears in both decompositions.)
Step 3 — Compare and cancel the identical segments
Since Area(semicircle on AB) = Area(sector AOB), using the two decompositions we get
Area(lune) + Area(circular segment) = Area(triangle OAB) + Area(circular segment).
Cancel the identical circular-segment term from both sides to obtain
Area(lune) = Area(triangle OAB).
Step 4 — (Optional) compute the triangle area numerically
Triangle OAB is isosceles right with legs OA = OB = R, so
Area(triangle OAB) = (1/2)·OA·OB = (1/2)·R·R = R^2/2.
Thus the lune has area R^2/2 as well. This proves the lune and the triangle have equal areas.
Why this argument is valid (one-sentence justification):
Both the semicircle on AB and the sector AOB include the identical curved region bounded by chord AB and the arc AB of the original circle, so subtracting that common piece leaves triangle = lune because the two whole areas are equal.
Teacher comment + rubric (Amy Chua "Tiger Mother" cadence — 100 words)
Good. You will do this precisely and without hand‑waving. Label the circle centre O and vertices A, B so OA = OB = R and ∠AOB = 90°. Compute areas: semicircle on AB = ½π(AB/2)^2 = πR^2/4, sector AOB = 90°/360°·πR^2 = πR^2/4. The sector = triangle OAB + the circular segment; the semicircle = the lune + the same circular segment. Cancel identical segments; the lune = triangle OAB. Show every algebraic step, diagram neat, units written, and provide justification. No shortcuts. Turn it in.
Rubric (short)
- Correctness of reasoning and result: 40% (lune = triangle)
- Completeness of steps and algebra: 25%
- Quality of diagram and labels: 15%
- Clarity of explanation and justifications: 20%
ACARA v9 mapping
This task maps to ACARA v9: Geometry and measurement — use geometric reasoning and area formulas; construct and interpret diagrams; present a proof-style argument. Suitable for middle secondary Years 7–9 reasoning strands (prove and justify results about shapes and areas).
Enrichment and extension
- Generalize: Investigate which other right triangles produce lunes whose areas equal some simple triangle or semicircle area. Study Hippocrates' other lunes.
- Coordinate proof: Place O at (0,0), A(R,0), B(0,R) and compute the lune area by integration (set up the x or y integral for the region inside semicircle and outside circle) to verify the same numeric answer.
- Constructive challenge: Draw semicircles on the two legs OA and OB (outside the triangle). Show algebraically that the sum of their areas equals the semicircle on AB (use a^2 + b^2 = c^2). Relate that to the partitioning of regions in the diagram.
- Play with dynamic geometry software (GeoGebra): animate R and watch the triangle and lune areas stay equal.
Helpful hint for working student: Always draw the chord AB, label the regions, and write each area decomposition explicitly before cancelling — that makes the argument clean and impossible to misinterpret.