I think We would need to know "how" the cornerers are rounded. Also, it isn't clear if the outer curve if circular, elliptical or otherwise (I'm assuming circular from the pizza analogy, but it should be stated).A picture doesn't say that.
edit:
Assuming we know the function of the rounded portion is a function r of theta, and that the outer portion has eccentricity 1, I would break the area into 3 pieces
From theta = 0 to alpha, alpha to 2*beta-alpha and finally from 2*beta- alpha to 2*beta. Alpha is the angle from the left edge to where the "rounding curve" feeds into the circle and Beta is half the angle sweeped out in the sector.
Your area would then be \(\displaystyle 2\int_0^{\alpha}r(\theta)d\theta + 2R(\beta-\alpha)\), where R is the radius of the large circle.
