Calculation of the area of a circle segment in cubic inches

I am trying to calculate the amount of displacement a barrel partially submerged has in cubic inches.

barrel is 20x30.

I am able to calculate the chord length and height at a given circle center angle but when attempting to convert to cubic inches, the values are too large.example: using barrel dimensions above and a radius of 10 and a center angle of 135, i get a chord length of 18.48, height of 6.17 and area of 82.45.

When i attempt to get the volume of the area by 82.45 x 30 = 2473.03 cubic inches.

2473/231 (cubic inches per gallon of water) = 10.71 gallons.

10.71 x 8.3 (#'s/gallon water) = 89.31 displacement pounds.

I believe the area of the segment doesn't represent square inches but I don't know why or what it represents instead. A full 30 gallon barrel submerged should only displace about 250 pounds. My calculations show a full barrel displacing about 340 pounds so an assumption I have made about the area calculation/cubic volume result is wrong and I don't know why.

The volume calculation takes the area of the segment here:

82.45 =((10^2)/2)*(2.36-SIN(2.36)) (2.36 is radians = 135 degrees)

The volume calculation takes the area, thought to be square inches, multiplied by the length of the barrel (30).

82.45 * 30 = 2573.63 cubic inches

Gallon calcullation

2573.63 / 231 = 10.71 (231 is # cubic inches per gallon water)

Pounds Calculator (# gallons x wt per gallon)

10.71 * 8.3 = 89.31 pounds displacent. (8.3 pounds per gallon of water.)