Make Your Valentine Proud

Assume that $n$ is the small square size

( Small square area - Cicle quarter area ) + ( Rectangle Area - Circle Quarter area ) + ( Rectangle Area - Semicircle Area ) $=91$

$(n^2 - (\frac{22}{7} \times \frac {1}{4} \times n^2 )) + (( n \times 2n )- (\frac{22}{7} \times \frac {1}{4} \times n^2 )) + (( n \times 2n )- (\frac{22}{7} \times \frac {1}{2} \times n^2 )) = 91$

$n^2 - \frac {11}{14} n^2 + 2n^2 - \frac {11}{14}n^2 + 2n^2 - \frac {11}{7} n^2 = 91$

$\frac{13}{7} n^2 = 91$

$n^2 = 49$

$n= 7$

Now we can conclude that,

Square Paper Area - Orange Area = Red Heart Area

$(3(n))^2 - 91 =$ Red Heart Area

$9 \times 7^2 - 91 =$ Red Heart Area

$441-91 =$ Red Heart Area

$\boxed {350} =$ Red Heart Area

Let $r$ to be the radius of the two red semi-circles, then $2r + r = 3r = \text{side of greatest square} \Rightarrow$ $\text{Surface of the greatest square} = 9r^2$

$\text{Surface of orange color} = r^2 + (4r^2 - \pi \cdot r^2) = 5r^2 - \frac{22}{7} \cdot r^2 =$ $\frac{13}{7} \cdot r^2 = 91 \Rightarrow r^2 = \frac{637}{13} = 49 \Rightarrow$ $\text{Surface or heart shape} = \text{Surface the greatest square} - \text{Surface of orange color} =$ $= 9 \cdot 49 - 91 = 441 - 91 = 350$

No indication from the problem statement that the answer will be decimal, we can safely assume the area of the square containing the heart is a perfect square and the heart-shaped area will be presumably an integer.

The orange area from visual estimation is about 1/3 to 1/4 of the heart. No way it's 1/2, 1/5 is kinda too small. 91 x 3 is 273. 91 x 4 is 364. The red area should be in around 273 to around 364 that adds to 91 to make a perfect square. Thus, area of the great square should be around 364 to around 455.

Perfect squares from 300: 324, 361, 400, 441, 484.

324 - 91 = 230 ish doesn't make sense, too small 361 - 91 = 250, less than 1/3, ehhhh ok 400 - 91 = 300ish, sounds good 441 - 91 = 350, sounds good 484 - 91 = 390, kinda too big

Try the 2 good ones. If failed, calculate it as written in the other solution :)

Cheers!