How far does the centre travel?
A hard-level "Length" problem.However many sides the polygon has, the corner arcs always make one full circle. "Perimeter + 2 × radius × π" and you're done.
GivenSide 8 cm square / circle radius 1 cm / π = 3.14
Hints
- It is the circle that moves, but what is asked is the path of its centre. The dotted line is that path.
- Split the path into two kinds. First the straight parts (red). Here the centre moves parallel to a side, by the same length as that side.
- The square has 4 sides of 8 cm each, so the straight parts total 8 × 4 = 32 cm.
- The other kind is the turns at the corners (blue). Going round a corner, the centre traces an arc of radius 1 cm centred on that vertex.
- The central angle of each arc equals the exterior angle of the square. One interior angle is 90°, so the exterior angle is 180 − 90 = 90°. All 4 corners are the same.
- 90° at 4 corners gives 90 × 4 = 360° — one full circle. So the 4 arcs together make exactly one circle of radius 1 cm.
- A circle's circumference is "diameter × π" = "2 × radius × π": 2 × 1 × 3.14 = 6.28 cm.
- Add the 32 cm of straight path and the 6.28 cm of corners. 32 + 6.28
Answer 38.28cm
More "Length" problems
- a + b + c = ?
- What is the side of the square?
- How long is MN?
- How long is the rope?
- How long is the pole's shadow?
- How deep was the water?