Problem
Slide pieces so the shaded part becomes one lump
A circle is cut into six triangles and six outer pieces, and half of each kind is shaded. We need the total shaded area.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — Neither an equilateral triangle nor a curved outer piece has an area a sixth-grader can work out. But the shading takes exactly half of each kind, and half of everything is half the circle -- so the pieces never have to be measured at all.
1STEP 1
Sort the circle into two kinds of piece
The circle is made of six congruent triangles inside the hexagon and six congruent pieces between the hexagon and the circle.
Twelve pieces, in two matching sets of six.
6.G.A.1Visualize Spatial Relationships2STEP 2
Count how many of each kind are shaded
Three of the six triangles are shaded, and three of the six outer pieces are.
6 ÷ 2 = 3
Exactly half of each kind.
6.G.A.1Change Focus Count The Complement3STEP 3
Slide the shaded pieces together
Half the triangles plus half the outer pieces is half of everything in the circle -- the shaded parts slide together into a semicircle.
The shaded area is half the circle.
7.G.B.4Visualize Spatial RelationshipsThe shaded pieces together are half the circle.
Why?
The circle is made of six triangles and six outer pieces, with nothing left over and nothing counted twice.
🧱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?
The six triangles are all the same size and so are the six outer pieces, because turning the picture a sixth of a turn carries each onto the next.
🧱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?
Exactly three of each kind are shaded, so the shading takes half of every kind and therefore half of everything.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
4STEP 4
Take half the circle
Work out the circle and halve it.
12 × 12 × 3.14 ÷ 2 = 226.08
The shaded parts come to 226.08 cm².
7.G.B.4Draw A DiagramAnswer
226.08 cm²
Check by symmetry: turning the picture by one sixth of a turn swaps the shaded and unshaded pieces exactly, so the two must have the same area, and together they are the whole circle.
Takeaway
If the shading takes half of every kind of piece, it takes half the whole.
- Sort the circle into two kinds of piece
- Count how many of each kind are shaded
- Slide the shaded pieces together
- Take half the circle