Halving the Perimeter of a Heart-Shaped Curve

angle ϕ

0.5236

show angle

A curve consists of two semicircles of radius 1 and one semicircle of radius 2. A chord through the origin and inclination

ϕ

divides the curve into two parts, red and blue. Which curve is longer?

The two parts have equal lengths. The blue part consists of a semicircle of length

π

: arc

DO

of length

π-2ϕ

and arc

AE

of length

2ϕ.

Together, these are

2π

. The length of the whole curve is

4π

. So the dashed line halves the perimeter.

[1] D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin Books, 1991, p. 30.

Izidor Hafner