Problem
Measure a figure in copies of a unit piece
A square is cut into the seven tangram pieces. The smallest triangle has area 1 in². Compare pieces instead of measuring lengths. Find the area of each of the other six pieces.
Reasoning · Grade 5-1 Equal-Area Transformation (1)
Your answer
Watch out Different shapes suggest different areas, but the square and the parallelogram match.
How to solve
Strategy Draw a Diagram — The fastest route is to draw a few extra cut lines onto the tangram picture until every one of the seven pieces has been chopped into copies of the small shaded triangle. Then no area formula is needed at all — each piece's area is simply the number of little triangles it contains. Six pieces means six small subproblems, each one 'how do I slice this piece into copies of the unit?', and each is easy because every tangram piece is built from that one triangle. With real tangram pieces you can do the whole thing by hand, laying the small triangle on each of the others and turning it. At the end the seven counts must add up to the whole square, which is a strong check on all six answers at once.
1STEP 1
Fix the unit and the size of the whole square
Take the small triangle as the unit.
square = 4 × 4 = 16, shaded triangle = 1/2 × 2 × 1 = 1
Area measured in copies of a chosen piece works exactly like area measured in unit squares: cover the shape with the piece, no gaps and no overlaps, and count. Here the chosen piece happens to be worth one square unit, which is a lucky bonus but not the point.
3.MD.C.5Draw A Diagram2STEP 2
The other small triangle is a straight copy
The other small triangle is also 1.
S2 = 1/2 × 2 × 1 = 1 in²
Every tangram has two identical small triangles; spotting the twin first costs nothing and gives you a second reference piece to lay on the others.
6.G.A.1Create A Physical Representation3STEP 3
The square is two unit triangles
The square piece is 2.
Q = 1 + 1 = 2 in²
One extra line turns an unfamiliar tilted square into two shapes you already know the area of — that is the whole method of decomposing a polygon.
6.G.A.1Identify Subproblems4STEP 4
The medium triangle is two unit triangles
The medium triangle is also 2.
M = 1 + 1 = 2 in²
In any right isosceles triangle, the line from the right angle to the middle of the longest side cuts it into two smaller right isosceles triangles of exactly half the size — the single fact that runs through this whole tangram.
6.G.A.1Identify Subproblems5STEP 5
The parallelogram is two unit triangles
The parallelogram is 2 as well.
P = 1 + 1 = 2 in²
Cutting any parallelogram along a diagonal gives two matching triangles, so once one of them is identified the other is free.
6.G.A.1Identify Subproblems6STEP 6
Each large triangle is four unit triangles
Each large triangle is 4.
L1 = 2 + 2 = 4 in², L2 = 4 in²
Building on a piece you have already measured saves work: you do not have to draw all four little triangles inside the big one, only see it as two mediums.
6.G.A.1Identify Subproblems7STEP 7
Check: the seven pieces must rebuild the whole square
The seven add back to 16, the whole square.
4+4+2+2+2+1+1=16
Because the pieces come from one square with no gaps or overlaps, their areas have to add back to the square's area — the single check that tests all six answers at once.
4.NBT.B.4Draw A DiagramThe seven pieces must rebuild the whole square, so the measured areas have to add back to it exactly.
Why?
The pieces cover the square with no gap and no overlap, so their areas put together are the square's area.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Moving a piece to lay it against the unit triangle keeps its area, so comparing pieces by stacking them is a fair measurement.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Answer
4, 4, 2, 2, 2, 1 in²
4 + 4 + 2 + 2 + 2 + 1 + 1 = 16
Every area is a whole number of square inches, which it must be, since each piece was cut into whole copies of the unit triangle. The order of the answers matches what the eye sees in the picture: the two biggest pieces are the large triangles at 4 each, the three middle-sized pieces (medium triangle, square, parallelogram) all come out equal at 2 — and they do look the same size — and the two small triangles are the smallest at 1. The strongest check is the total: 4+4+2+2+2+1+1=16, exactly the area of the square, which on the grid used is 4 × 4=16 square units. A second reading of the same picture confirms it: the diagonal (0,0)–(4,4) splits the square into two halves of 8 each, and it happens to be a cut line, so the pieces sort cleanly. Below it lie exactly the two large triangles, 4+4=8; above it lie the medium triangle, the square, the parallelogram and both small triangles, 2+2+2+1+1=8.
Takeaway
Pick one little piece as your unit, and every other piece's area is just how many copies of it fit inside — no formulas needed.
- Fix the unit and the size of the whole square
- The other small triangle is a straight copy
- The square is two unit triangles
- The medium triangle is two unit triangles
- The parallelogram is two unit triangles
- Each large triangle is four unit triangles
- Check: the seven pieces must rebuild the whole square