The dip to litres formula

The shape

An underground fuel tank lies on its side: a cylinder with flat ends. Looking at one end, the fuel fills the bottom of a circle up to the dip. That shape is a circular segment. The litres are the segment’s area times the tank’s length.

The formula

With r the inside radius, h the dip and L the length, all in centimetres:

segment area = r² · acos((r − h) / r) − (r − h) · √(2rh − h²)     (cm²)
litres       = segment area × L ÷ 1000

The first part, r² · acos(…), is the slice of the circle the segment sits in. The second part takes off the triangle between the slice’s edges and the centre.

Worked example: BPCL 20 KL, 210 cm across

r = 105 cm, L = 600 cm, h = 142.5 cm.

1. r − h = 105 − 142.5 = −37.5
2. (r − h) / r = −0.357143; acos(−0.357143) = 1.936004 radians
3. r² · acos = 11,025 × 1.936004 = 21,344.44
4. 2rh − h² = 29,925 − 20,306.25 = 9,618.75; √ = 98.0752
5. (r − h) · √ = −37.5 × 98.0752 = −3,677.82
6. area = 21,344.44 − (−3,677.82) = 25,022.26 cm²
7. litres = 25,022.26 × 600 ÷ 1000 = 15,013.36 L

This is the figure the app shows for the same tank and dip. The calculator does it for any tank.

Deadwood

IndianOil prints a deadwood figure on its charts: litres taken up by fittings inside the tank. Take it off in proportion:

litres = cylinder litres × (1 − deadwood ÷ full cylinder litres)

For the IOCL 20 KL tank the deadwood is 18 litres.