Area Maze
Every hidden length is one trade away.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: Every area is a side times a side, so every known area trades into one missing length.
L0 · If two rectangles share a wall AND one rectangle's area + that-side is known, then that side's length = area ÷ side; the SHARED wall is the same in both.
A reading-corner mat is made from two rectangles in a single row that share a vertical wall. The left rectangle has area 12 square centimeters and is 3 centimeters wide. The whole row is 8 centimeters wide. Find the area of the shaded rectangle on the right.
L1 · If a length recovered in one place is needed elsewhere, then walk it across the figure, adding or subtracting the known offsets between rectangles.
A welcome sign is built from two connected rectangles in a step shape. The top rectangle has area 56 square centimeters and is 7 centimeters tall. The shaded rectangle below sticks out 5 centimeters to the left of the top and is 4 centimeters short of the top on the right. The shaded rectangle is 6 centimeters tall. Find its area.
L2 · If two rectangles have EQUAL areas AND share a side, then their OTHER sides are equal too.
A comfort blanket strip is made from three rectangles stacked vertically that all share the same width. The top has area 21 square centimeters. The bottom has area 21 square centimeters. The full strip is 10 centimeters tall. A side rectangle of area 36 square centimeters and width 9 centimeters is attached to the right, sharing the strip's full height. Find the area of the shaded middle rectangle.
L3 · If two (or more) adjacent rectangles share a common side, then (sum of areas) ÷ (shared side) = sum of the other sides.
A welcome banner has three rectangles in a row at the bottom, with areas 12, 16, and 20 sq cm. They all share the same height. A top cap rectangle sits on the first two bottom rectangles. The last two bottom rectangles together are 7 centimeters wide. The cap is 5 centimeters tall. Find the area of the cap.
L4 · If two adjacent rectangles share a side AND you're given the DIFFERENCE in their areas (and the difference in their other sides), then shared side = (area difference) ÷ (side difference).
A community mural frame is split into three rectangles in a row, all sharing the same height. The left has area 12 square centimeters. The right has area 24 square centimeters. The left two rectangles together span 8 centimeters in width. The right two rectangles together span 11 centimeters in width. Find the area of the shaded middle rectangle.
L5 · Pick the right principle at each step — L0–L4 are the toolbox, and L5 puzzles need 2–3 chained correctly.
A relief map display has three rows of rectangles arranged in offset columns. The top row covers two columns with total area 36 square centimeters over a combined width of 9 centimeters. The middle row has three side-by-side rectangles with areas 18 on the left, 30 on the right, and an unknown middle. The bottom row covers two different columns with total area 21 square centimeters over a combined width of 7 centimeters. Find the total area of the whole middle column from top to bottom.
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
L0 · Shared Side
Peek the trick — Shared Side
Two rectangles glued along one wall must share that wall's length. Recover the hidden length on one side using area divided by the known side, then carry that length straight to the other rectangle.
L1 · Carry the Length
Peek the trick — Carry the Length
When two rectangles are offset rather than aligned, the length you already paid for keeps working. Walk it across the figure, adding the visible overhangs and subtracting the visible underhangs.
L2 · Equal Areas, Equal Sides
Peek the trick — Equal Areas, Equal Sides
Two rectangles with the same area and a shared side must match on their other side as well. Equal areas plus equal width force equal heights, so the leftover space splits cleanly with no algebra.
L3 · Group and Divide
Peek the trick — Group and Divide
When several rectangles share one height, group their areas and divide by the combined width. The whole group acts like one rectangle, so a single division recovers the shared dimension.
L4 · Difference Trade
Peek the trick — Difference Trade
Do not fight the unknown — subtract it away. The difference of two areas divided by the difference of their visible spans gives the shared dimension, because the unknown cancels in the subtraction.
L5 · Area Detective
Peek the trick — Area Detective
No single rule cracks the figure — chain two or three. Apply each tool where the picture asks for it, passing the recovered length forward until the final area falls out.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
