Problem
Area of pieces that split a rectangle
A square has an area of 64 square inches. Each side is marked into 4 equal parts. One marked point on each side is joined in order. Find the area of the quadrilateral formed.
Reasoning · Grade 5-1 Area of Figures
Your answer
Watch out The tilted quadrilateral looks hard to measure, but each side is a rectangle's diagonal.
How to solve
Strategy Draw a Diagram — A tilted quadrilateral has no formula I can use directly, so I add two helper lines to the drawing: the vertical line through the top and bottom chosen points, and the horizontal line through the left and right chosen points. Those two lines cut the square into four rectangles, and — this is the whole trick — each side of the quadrilateral turns out to be a diagonal of one of those rectangles. A diagonal splits a rectangle into two equal halves, so each rectangle is exactly half shaded, and adding them up settles the whole figure. As a cross-check I will change focus and instead measure the four leftover corner triangles and subtract them from 64.
1STEP 1
Get the side length and the spacing of the marks
The side is 8 and the marks are 2 apart.
8 × 8 = 64 → side = 8 in; 8 ÷ 4 = 2 in
Every length in the picture is a whole number of 2-inch steps, so once the step size is known, any distance in the figure can be read off by counting marks.
3.MD.C.7Draw A Diagram2STEP 2
Pin down where the four chosen points are
Pin down where the four chosen points sit.
Noticing that the top and bottom points line up vertically, and the left and right points line up horizontally, is what makes the helper lines land exactly on the chosen points.
6.G.A.1Draw A Diagram3STEP 3
Draw the two helper lines and get four rectangles
Two helper lines make four rectangles.
2 × 6 = 12, 6 × 6 = 36, 2 × 2 = 4, 6 × 2 = 12; 12+36+4+12 = 64
Checking that the four rectangle areas add back to 64 confirms the helper lines were placed correctly before any harder reasoning is built on them.
3.MD.C.7Identify Subproblems4STEP 4
See that each side of the quadrilateral is a rectangle's diagonal
Each side of the quadrilateral is one diagonal.
This is the payoff of drawing the helper lines through the chosen points — the awkward slanted sides stop being awkward and become ordinary rectangle diagonals.
6.G.A.1Draw A Diagram5STEP 5
Use the fact that a diagonal halves a rectangle
A diagonal splits a rectangle in half.
12/2 + 36/2 + 4/2 + 12/2 = 6+18+2+6
A triangle sitting in a rectangle and sharing its base and height is always half the rectangle, which is the one geometric fact this whole problem turns on.
6.G.A.1Identify SubproblemsA diagonal cuts a rectangle into two pieces of exactly the same area, so each corner triangle is half its rectangle.
Why?
Turning one half a half-turn about the rectangle's centre lays it exactly onto the other half, so the two must be equal.
🧱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?
Two equal pieces filling the rectangle with no gap and no overlap means each is one of two equal shares of it.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
6STEP 6
Add up the halves
The halves add to 32.
6+18+2+6 = 32 = 64 ÷ 2
Because every single rectangle is half shaded, the halves add without any special cases, and the answer comes out as exactly half the square no matter how the pieces were sized.
6.G.A.1Identify Subproblems7STEP 7
Cross-check by measuring what is left over instead
Measuring the leftovers instead also gives 32.
64 - ((2 × 6)/2+(6 × 6)/2+(6 × 2)/2+(2 × 2)/2) = 64-(6+18+6+2) = 64-32 = 32
The leftover pieces are plain right triangles with horizontal and vertical legs, so each one is half of an easy rectangle — a completely different route to the same number.
6.G.A.1Change Focus Count The ComplementAnswer
32 in²
64 ÷ 2 = 32
The units are right: lengths were in inches, every area came from an inch times an inch, and the answer is in square inches. The magnitude is right too — the quadrilateral clearly fills a good deal of the square but not all of it, and 32 in² is exactly half of 64 in², which matches how the shading looks in the figure. The two methods agree: adding four half-rectangles gave 6+18+2+6 = 32, and subtracting the four corner triangles gave 64-32 = 32. As a final consistency check, the shaded and unshaded parts came out equal at 32 each, and they sum to the full 64 in².
Takeaway
Draw lines through the chosen points to split the square into four rectangles — each side of the shape is a diagonal, so exactly half of every rectangle is shaded and the answer is half the square.
- Get the side length and the spacing of the marks
- Pin down where the four chosen points are
- Draw the two helper lines and get four rectangles
- See that each side of the quadrilateral is a rectangle's diagonal
- Use the fact that a diagonal halves a rectangle
- Add up the halves
- Cross-check by measuring what is left over instead