# How to find area of intersection of disk and cone?

I'm trying to calculate the area of intersection of a cone and a disk which is a circular slice of the cone rotated through a particular angle about the apex of the cone. I've defined the cone using a set of inequalities with ImplicitRegion and have found a parametrization for the circle. I'm confident I have found the right equations because plotting RegionUnion produces what I expect to see, for example for the disk rotated through 1 degree:

R = 1.4;
d = 30;
T = Pi/180;
cone = ImplicitRegion[{x^2 + y^2 <= (z Tan[R/d])^2,0 <= z < 1.2*d}, {x, y, z}];
disk = ParametricRegion[{d Sin[T] - r Cos[T] Cos[t], r Sin[t],d Cos[T] - r Cos[t] Sin[T]}, {{r, 0, 1.42}, {t, 0, 2 Pi}}];
Region[cone]
Region[disk]
Region[RegionUnion[cone, disk]]


However, I can't get it to plot the intersection or calculate the area of the intersection:

Region[RegionIntersection[cone, disk]]


produces the output

and asking it for the area of intersection,

Area[RegionIntersection[cone, disk]]


gives the output:

Can anyone help me figure out why this isn't working and how to calculate the area of intersection?

In[7]:= RegionMeasure[RegionIntersection[cone, disk]]