Problem
Move a figure to make its area easy
A circle inside a big square touches all four sides. A smaller square sits in the circle with its corners on it. The shaded small square covers 8 square inches. Find the area of the big square.
Reasoning · Grade 5-1 Equal-Area Transformation (1)
Your answer
Watch out The radius seems needed, but a simple turn solves it without the circle.
How to solve
Strategy Create a Physical Representation — The shaded square is in an awkward place: it sits inside the circle, and nothing lines it up with the big square. But its area does not care where it is - if I cut it out and turn it 45 degrees, it still covers 8 square inches. Turning it 45 degrees about the centre swings its corners onto the four points where the circle touches the big square, which are the midpoints of the big square's sides. In that position the two squares share a centre and the picture cuts cleanly into congruent triangles that I can simply count. This 'move it somewhere convenient, then count pieces' idea is exactly the equal-area transformation the unit is about.
1STEP 1
See that the circle touches each side at its midpoint
The circle touches each side at its midpoint.
diameter of circle = side of the big square
A square has four lines of symmetry through its centre, and each one hits a side at its midpoint - so a circle that fits snugly can only touch there.
4.G.A.2Draw A Diagram2STEP 2
Turn the shaded square 45 degrees about the centre
Turn the small square 45 degrees.
area after turning = area before turning = 8 in²
This is the one fact the whole problem rests on: a rotation moves a figure without stretching or squashing it, so lengths and area are exactly the same afterwards. You can check it by tracing the square on paper and spinning the tracing.
8.G.A.1Create A Physical RepresentationTurning the shaded square 45 degrees about the centre keeps its area while making it line up with the big square.
Why?
A turn lays the square onto a copy of itself, so every side keeps its length and the area travels along unchanged.
🧱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).
Why?
In its new position the shaded square is exactly some of the identical triangles the big square splits into, so counting them measures it.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Cut the big square into 8 identical triangles
Cut the big square into 8 identical triangles.
4 small squares × 2 = 8 identical right triangles
Breaking a shape into copies of one small piece turns an area question into a counting question - and a diagonal always splits a square into two matching halves.
6.G.A.1Identify Subproblems4STEP 4
Count how many triangles the shaded square covers
The turned square covers 4 of them.
4/8=1/2
Areas add up: if eight identical pieces make the big square and four of them make the shaded one, the shaded one must be half - no lengths needed at all.
3.MD.C.7Visualize Spatial Relationships5STEP 5
Double the shaded area
So the big square is 8 × 2 = 16.
8 × 2 = 16 in²
Once the picture says 'the big one is two of the small one', the arithmetic is a single doubling that any Grade 3 student can do.
6.G.A.1Identify SubproblemsAnswer
16 in²
8 × 2 = 16
The answer is in square inches, the right unit for an area, and 16 square inches is bigger than the 8 square inches of the shaded square - which it must be, since the shaded square sits strictly inside the big one. It is also a believable amount bigger: the circle fills most of the big square and the shaded square fills a good part of the circle, so a factor of exactly 2 is sensible, not a factor of 10. A coordinate check agrees: put the big square from (0,0) to (4,4), so its area is 16; the circle then has centre (2,2) and radius 2, and the inscribed square with horizontal sides has corners at (2 - sqrt(2), 2 - sqrt(2)) and so on, giving a side of 2 x sqrt(2) and an area of 8 - exactly half of 16.
Takeaway
Give the shaded square a 45-degree turn - it keeps its area but lands in a spot where the big square is obviously just two of it.
- See that the circle touches each side at its midpoint
- Turn the shaded square 45 degrees about the centre
- Cut the big square into 8 identical triangles
- Count how many triangles the shaded square covers
- Double the shaded area