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Calculating Horizontal Cylinder Volume
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To find the volume of a horizontal cylinder, we must
1) Calculate the entire area of one cylinder "end".
2) Calculate the segment area - (the blue area) by:
subtracting triangle Area AOB from the sector area
(sector area is the blue area plus the red area)
3) Divide segment area by area of the cylinder end.
4) Multiply this by the entire volume of the cylinder.
Let's work on a problem.
A horizontal cylinder has a length of 200 centimeters and a diameter of 100 centimeters.
It is partially filled to a depth of 20 centimeters.
What is the volume when filled to this depth?
First, we'll make some calculations to be used later:
Total Cylinder Volume = π • r² • length = π • 2,500 • 200 = 1,570,796 cm³
Cylinder "End" Area = π • r² = π • 50 * 50 = 7,853.98 cm²
We need to find the area of the segment (the blue area).
To do this we must first calculate the area of the sector, (red and blue areas),
then subtract the red triangle area.
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Line OE = radius - depth
Line OA, the tank's radius is 50 cm and line OE is 30 cm.
arc cosine ∠ AOE = OE ÷ OA = 30 ÷ 50 = .6
∠ AOE = 53.13°
∠ AOB = 53.13° * 2 = 106.26°
Sector Area = (106.26 ÷ 360) • 7,853.98 cm² = 2,318.23 cm²
Next we calculate the red triangle area.
Line AE2 = radius2 -Line OE2
Line AE2 = 502 -302
Line AE2 = 2,500 -900
Line AE2 = 1,600
Line AE = 40
Area of Red Triangle = Line AE • Line OE
Area of Red Triangle = 40 • 30 = 1,200 cm²
Finally, we get to calculate the segment area.
Segment Area = Sector Area -Red Triangle Area
Segment Area = 2,318.23 cm² - 1,200.00 cm²
Segment Area = 1,118.23 cm²
Volume at 20 cm = Segment Area ÷ "End" Area • Total Volume
Volume at 20 cm = 1,118.23 ÷ 7,853.98 • 1,570,796 cm³ =
Volume at 20 cm = 0.1423779213 • 1,570,796 cm³ =
Volume at 20 cm = 223,647 cm³
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