Problem
Measure straight parts as sides, curved parts as arcs
Two arcs are drawn inside a square and a region between them is shaded. We need the length all the way round that region.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — A perimeter is a walk round the edge, so trace it and name each stretch as you go. Two of the three stretches are arcs; the third is a side of the square. The bottom side is not on it at all -- the semicircle sits on top of it and takes its place.
1STEP 1
Trace the boundary
Starting at the top-left corner, go down the left side, along the semicircular arc to the bottom-right corner, then back up the big arc to where you started.
One straight stretch and two curved ones.
7.G.B.4Visualize Spatial RelationshipsThe square's bottom side is not part of the shaded part's boundary.
Why?
The semicircle sits on that whole side and is cut away, so along the bottom the shaded part stops at the arc, not at the side.
🧱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?
A perimeter follows wherever the region actually ends, so it follows the arc there and the side never appears.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Measure the quarter arc
It is a quarter of a circle whose radius is the square's side.
16 × 2 × 3.14 ÷ 4 = 25.12
The big arc is 25.12 cm.
7.G.B.4Draw A Diagram3STEP 3
Measure the semicircular arc
Its diameter is the bottom side, so its radius is half of that, and a semicircle is half a circumference.
8 × 2 × 3.14 ÷ 2 = 25.12
The small arc is also 25.12 cm.
7.G.B.4Draw A Diagram4STEP 4
Add the three stretches
The straight stretch is the square's left side.
16 + 25.12 + 25.12 = 66.24
The perimeter is 66.24 cm.
7.G.B.4Identify SubproblemsAnswer
66.24 cm
The two arcs come out the same length, which they should: doubling the radius and halving the fraction of a circle cancel exactly.
Takeaway
A perimeter follows the actual edge -- trace it and name each stretch.
- Trace the boundary
- Measure the quarter arc
- Measure the semicircular arc
- Add the three stretches