Do you plan to keep your cow's calf for butcher? If so, that's a hay expense you'll have to factor in. If not, selling the weaned calf could bring in some hay money! Then again, do you have means to breed her again (trailer) or someone to AI her? Just things to keep in mind......
With that...
You can absolutely keep a cow on half an acre. Sounds like you're just planning to feed hay, and that's what folks are talking about. A half acre won't have enough feed to sustain a cow, but it's plenty of physical space for her. I keep two horses in a small pen on my acre, they'd be thrilled to...
So what's the green area? Lawn? Around here "lawn" is just another word for pasture! Fence that in and put an animal on it!
I also live on one acre, but have a very long, narrow property. I'm envious of your nice square!