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# Finding the volume of a rugby ball (or American football) | IB Maths Resources from Intermathematics if you’re a teacher, also visit my new site: intermathematics.com for over 2000 pages of ib maths teaching resource pdfs.

Reading: What is the volume of a football

find the volume of a rugby ball (elongated spheroid)

With the rugby world cup currently underway, I thought I’d try to calculate the volume of a rugby ball using some calculations. this method works similarly for American football and Australian rules football. The approach is to consider the rugby ball as an ellipse rotated 360 degrees about the x-axis to create a volume of revolution. we can find the equation of an ellipse centered at (0,0) simply by looking at the x and y intercepts. an ellipse with y-intercept (0,b) and x-intercept (a,0) will have the equation:  so for our rugby ball with a horizontal (vertex) “radius” of 14.2 cm and a vertical (co-vertex) “radius” of 8.67 cm, you will have the equation: We can see that when we plot this ellipse we get an equation that closely resembles the shape of our rugby ball: so now we can find the volume of revolution using the following formula: but we can simplify things by starting the rotation at x = 0 to find half the volume, before doubling our answer. therefore:

reordering our equation of the ellipse formula we get: therefore we have the following integration:   so our rugby ball has a volume of around 4.5 litres. we can compare this to the volume of a football (soccer ball), which has a radius of around 10.5 cm, hence a volume of around 4800 cubic centimeters.

We can find the overall volume of any rugby ball (mathematically defined as an elongated spheroid) by the following generalization:       We can see that this is closely related to the formula for the volume of a sphere, which makes sense since the elongated spheroid behaves like a sphere deformed on its axes. our elongated spheroid has “radii” b, b, and a; therefore, r cubed in the sphere formula becomes b squared a.

elongated spheroids in nature

The image above [wiki image nasa] is of the Crab Nebula, a distant supernova remnant about 6500 light-years away. the shape of the crab nebula is described as an elongated spheroid.

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