Circles and perimeter

Geometry Level 2

What is the perimeter of the above figure?

25 π 25\pi 20 π 20\pi 36 π 36\pi 6 π 6\pi

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

Marta Reece
Jan 21, 2018

The centers A B C D ABCD form a square with side 6 6 .

Triangles A B D , B C E ABD, BCE , etc. are equilateral.

∡ D B E = 36 0 ∘ − 6 0 ∘ − 9 0 ∘ − 6 0 ∘ = 15 0 ∘ = 5 π 6 \measuredangle DBE=360^\circ-60^\circ-90^\circ-60^\circ=150^\circ=\frac{5\pi}6

Red arc = r ⋅ 4 ⋅ ∡ D B E = 6 ⋅ 4 ⋅ 5 π 6 = 20 π =r\cdot4\cdot\measuredangle DBE=6\cdot4\cdot\frac{5\pi}6=20\pi

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you for sharing a nice and logical solution.

Hana Wehbi - 3 years, 4 months ago
Edwin Gray
Feb 7, 2019

Clearly, each circle's red perimeter is less than 1/2 the circle. If 1/2, the answer would be 24 pi, So the only alternative is 20 pi, no thinking required. Ed Gray

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