Me Want a Heart Limb!

Let $r_2 = kr_1$ , such that $k > 1$ . The total area of the heart is $\begin{array}{rl} A_{\text{total}} &= A_{r_1} + A_{r_2} + A_{\text{bottom}}\\ &= \dfrac{\pi}{2} \left(r_1\right)^2 + \dfrac{\pi}{2} \left(r_2\right)^2 + \dfrac{\pi}{2} \left(r_1 + r_2\right)^2\\ &= \dfrac{\pi}{2}\left(\left(r_1\right)^2 + \left(r_2\right)^2 + \left(r_1\right)^2 + \left(r_2\right)^2 + 2r_1r_2\right)\\ &= \dfrac{\pi}{2}\left(2\left(r_1\right)^2 + 2\left(r_2\right)^2 + 2r_1r_2\right)\\ &= \pi \left(\left(r_1\right)^2 + \left(r_2\right)^2 + r_1r_2\right)\\ &= \pi \left(\left(r_1\right)^2 + k^2 \left(r_1\right)^2 + k\left(r_1\right)^2\right)\\ &= \pi \left(r_1\right)^2\left(1 + k + k^2\right) \end{array}$ The halved area is $A_{\text{halved}} = \dfrac{\pi}{2}\left(r_1\right)^2\left(1 + k + k^2\right)$ We want the new area to be half the original heart's area. In that case, the circular area of the cut must be congruent to the new area. So $\begin{array}{rl} \dfrac{\pi}{2}\left(r_1\right)^2\left(1 + k + k^2\right) &= \pi k^2 \left(r_1\right)^2\\ \dfrac{\pi}{2}\left(r_1\right)^2\left(1 + k - k^2\right) &= 0\\ 1 + k - k^2 &= 0\\ k^2 - k - 1 &= 0\\ k &= \dfrac{1 \pm \sqrt{1 + 4}}{2}\\ &= \dfrac{1 \pm \sqrt{5}}{2} \end{array}$ Thus, since $k > 1$ , the answer is $k = \boxed{\dfrac{1 + \sqrt{5}}{2}}$

Why can't $k$ be negative? Both possible solutions are shown in the graphic above, having in common the green semi-circle of radius $1$ .

In any case, it's a pretty picture.