Volume & Surface Area of an Elliptic Cylinder: Formulas and Examples
An elliptic cylinder is one whose horizontal cross-sections are ellipses or ovals. To find the surface area and volume of an elliptically shaped cylinder, you need to know the height of the cylinder, the length of the major ellipse axis and the length of the minor ellipse axis. With these three measurements, you can plug them into an equation for the surface area and an equation for the volume.
The volume formula for an elliptical cylinder is exact, but the surface area formula is an approximation since it is based on the perimeter of an ellipse, for which there is no closed-form formula. Along with the formulas are some solved example problems.
Elliptic Cylinder Volume Formula
Like circular cylinders and other 3-D shapes in the prism family, the volume of an elliptic cylinder is the area of the base times the height.
If the base is an ellipse with axis lengths of A and B, then the area of the base is πAB/4. If the height of the cylinder from end to end is H, then the volume is given by the equation
Volume = πABH/4.
If A = B, the base is a circle and A and B are the diameter lengths. The volume formula above reduces to (π/4)HD^2, the familiar formula for the volume of a circular cylinder with a diameter of D and a height of H.
Surface Area of an Elliptic Cylinder
The lateral surface area of an elliptic cylinder is the area of the vertical region between the elliptical base and top. The total surface area is the lateral surface area plus the areas of the top and bottom of the cylinder.
To find the lateral surface area, you multiply the perimeter or circumference by the height. Unfortnately, there is no closed-form expression for the circumference of an ellipse with a length and width of A and B, however, there are some approximation formulas. One is
(π/2)(A+B)[1 + Q2/4 + Q4/64 + Q6/256]
where Q = (A-B)/(A+B)
The lateral surface area and total surface area of an elliptic cylinder are then given by the formulas
Lateral Surface Area ≈
(πH/2)(A+B)[1 + Q2/4 + Q4/64 + Q6/256]
Total Surface Area ≈
(πAB/2) + (πH/2)(A+B)[1 + Q2/4 + Q4/64 + Q6/256]
Shaker oval boxes, sometimes simply called Shaker boxes, are hand-crafted wooden boxes with elliptical bases made in the style of Shaker furniture. The sides are made with a single piece of thin wood curled into an oval shape, with the ends joined together with a "finger" seam.
If a Shaker box has a length of 10 inches, width of 6.5 inches, ad a height of 5.5 inches, what is its volume in cubic inches?
For this problem we have A = 10, B = 6.5, and H = 5.5. Plugging these variable values into the elliptical cylinder volume equation gives us
Volume = πABH/4
= 280.78 cubic inches.
A tank is in the shape of an elliptical cylinder. The distance between the elliptical ends is 1.24 meters. The longer axis of the ellipse is 0.99 meters, and the shorter axis is 0.81 meters. What is its approximate lateral surface area of the tank?
In this problem we have A = 0.99, B = 0.81, and H = 1.24. First we compute the intermediate variable Q:
Q = (A-B)/(A+B) = 0.18/1.8 = 0.1 = 1/10
Now we can compute lateral surface area with the formula given above. This gives us
surface area ≈ (πH/2)(A+B)[1 + Q2/4 + Q4/64 + Q6/256]
= (π*0.62)(1.8)[1 + 1/400 + 1/640000 + 1/256000000]
≈ 3.515 square meters.
What if the base is not a perfect ellipse?
In plane geometry, the term "oval" is more general than "ellipse," so if the cylinder's base is an oval but not a true ellipse, then the volume and surface area formulas above will not be as accurate. Sometimes the shape called a "stadium" is called an oval. A stadium can be formed by taking a rectangle and attaching two half-circles to opposite sides, as in the geometric diagram below.
Diagram of a Stadium, Sometimes Called an "Oval"
For an oval cylinder of height H, whose base is a stadium with a length of A and a width of B, the formulas for volume and surface area are
- Volume = (πH/4)B^2 + BH(A-B)
- Lateral Surface Area = H(πB + 2A - 2B)
- Total Surface Area = H(πB + 2A - 2B) + (π/2)B^2 + 2B(A-B)
A container is in the shape of an oval cylinder where the base is a stadium curve rather than a true ellipse. The length of the base is 12 inches and the width is 8 inches. If the height of the container is 6 inches, what is its capacity in cubic inches?
For this problem, we have A = 12, B = 8, and H = 6. We need to use the volume formula given in the preceding section since the base is a stadium rather than an ellipse. The volume formula for this shape gives us
Volume = (πH/4)B^2 + BH(A-B)
= (6π/4)8^2 + 8*6(12-8)
= 96π + 192
≈ 493.5929 cubic inches