Image Source: http://resources3.news.com.au
The Melbourne Cricket Ground “MCG” is the undisputed home of AFL Football. At the end of season Grand Final it can accommodate up to 100,000 cheering fans.
The MCG actually comes in at Number 10 on the list of the world’s largest stadiums.
The MCG is used for Australian Football in winter and Cricket matches in summer.
An amazing video shows how they remove the Cricket pitches for the start of the AFL Football season, or any other special events at the MCG like music concerts and rugby matches.
The mathematics and Engineering used in designing the machine which moves the MCG wickets is world class, and Australia is a world leader in this area.
The turf moving machine needs to do the following:
1. Be able to evenly lift a large and massively heavy volume of stadium turf
2. Be strong enough to transport this very heavy load safely
3. Have giant wheels that evenly spread the weight of the load, and don’t leave giant tyre tracks across the rest of the field.
4. Be able to be retracted and unassembled for either storage inside the stadium, or transportation to other locations.
Here is the video which shows this amazing machine in action.
The following video gives an interesting six minute “Underground Tour” of the MCG during the Cricket Season.
Australian Football Stadiums
Apart from the MCG, there are several other stadiums around Australia where Football is played. Here are a few of the main ones.
ANZ Stadium, formerly Telstra Stadium and originally known as Stadium Australia, was built for the Sydney 2000 Olympic Games.
Four Boeing 747s would fit side-by-side under the span of the main arches of the grandstands.
The original capacity was 110 000, however post-olympics renovations has reduced this to 83 000.
Image Source: http://resources2.news.com.au
In Queensland, the main football stadium is the “Gabba” which is named after the Brisbane suburb it is located in. It’s seating capacity is only
42 000, and it is home to the Brisbane Lions football team.
Image Source: http://bottblog.dailymail.co.uk
In South Australia the main football ground is AAMI stadium in Adelaide, with a seating capacity of 51 500.
Image Source: http://media.au.timeout.com
Subiaco oval, officially known as Paterson’s Stadium, is located at Perth in Western Australia. It has a seating capacity of 43 000.
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The following table provides the size and capacity of each football ground.
Data Source: http://www.austadiums.com
An excellent website for getting very detailed information on all of Australia’s sports stadiums can be found at the following link.
Area and Perimeter of Football Grounds
Now that we have all of the data we we need, let’s get onto the mathematics of this post.
We know that the MCG is the tenth largest sports stadium in the world, but how much larger is it than the other Australian Football grounds ?
Our mathematics task is to calculate the Area and Perimeter of the MCG’s “oval”, (or “Ellipse”), shaped playing surface, and compare it with the size of some of the other Australian Football grounds.
First we need to know what the formulas are for the Perimeter of an Ellipse, and the Area of an Ellipse.
Area of an Ellipse
The area of an oval shaped “ellipse” can be calculated using the following mathematical formula.
In the above example we used 3.142 as our approximate value of Pi.
Area of the MCG Oval
As an example of the Area of an Ellipse, let’s calculate the Area of the MCG.
(See previous Data Table in this post for the dimensions of the MCG).
The Boundary to Boundary Length is the shorter dimension value of 141 m.
To get the “Short Radius” Length, we need to divide 141 by 2.
This gives a result of “a” = 70.5
The Goal to Goal Length is 160m.
To get the “Long Radius” Length, we need to divide 160 by 2.
This gives a result of “b” = 80
Area of MCG = Pi x 70.5 x 80 (where Pi = 3.142)
Area of MCG = 3.142 x 70.5 x 80
Area of MCG Oval = 17 720 square meters.
Perimeter of an Ellipse
The Perimeter of an oval shaped “ellipse” can be calculated using the following mathematical formula.
In the above example we used 3.142 as our approximate value of Pi.
Calculating the Perimeter of an ellipse if far more involved than calculating area.
What we suggest doing, is breaking the process down into the following series of steps.
Step 1. Halve the boundary to boundary shorter distance to work out “a”
Step 2. Halve the goal to goal longer distance to work out “b”
Step 3. Work out (a x a + b x b)
Step 4. Divide the Step 3 answer by 2.
Step 5. Work out the Square Root of the Step 4 answer.
Step 6. Multiply the Step 5 answer by 3.142
Step 7. Multiply the Step 6 answer by 2.
The answer to Step 7 is the final Perimeter Length value.
Perimeter of the MCG
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The dimensions of the MCG are 160 m x 141 m.
Following the previous seven step process we obtained the following answers.
Step 1. “a” = 141 divided by 2 = 70.5 m.
Step 2. “b” = 160 divided by 2 = 80 m.
Step 3. (a x a + b x b) = (70.5 x 70.5 + 80 x 80) = 11 370.25
Step 4. 11 370.25 divided by 2 = 5685.125
Step 5. Square Root of 5685.125 = 75.4
Step 6. 75.4 multiplied by 3.142 = 236.906
Step 7. 236.906 multiplied by 2 = 473.8
So the total length covered if we go all the away around the edge of the MCG oval is 474 meters.
The Perimeter of the MCG is 474 meters.
Evacuating the MCG Crowd onto the Oval
Image Source: http://ih2.redbubble.net
We know that on Grand Final day there are about 100 000 people at the MCG.
We also know that the Area of the MCG oval is 17 720 square meters.
Now let’s say there was an emergency during the Grand Final.
Could all of the crowd to come down from their seats and stand on the oval area?
Area per Person = 17 720 square meters divided by 100 000 people
Area per Person = 0.1772 square meters per person.
This would mean each person has to stand in area that is only 0.400 m by 0.443 m in size.
This area in cm would be 40 cm x 44 cm.
This is not enough room, because an Adult person has shoes that are around 35cm long, and shoulder width of around around 60cm.
Absolute Minimum Area per person, (with everyone jam packed together like sardines), would be 0.35 x 0.60 = 0.21 square meters.
To stand comfortable and have room for their bags, to do texting, and to sit down, a person would probably need nearly a whole square meter.
Therefore the 17 720 square meter area of the MCG oval could comfortably accommodate 17 720 people.
It is not possible to fit a crowd of 100 000 people onto the MCG playing area.
Area of the MCG including Grandstands
There is an interesting free online application from “Daft Logic” that uses the Google Maps Distance Calculator to find out the distance between two or more points anywhere on the earth. In other words, the distance from Point A to Point B.
The link to this free app is as follows:
We tried this app out at Passy World, and measured the total “a” and “b” distances across the MCG including the Grandstands.
The “a” distance was around 270 m divided by 2 = 135m.
The “b” distance was around 282 m divided by 2 = 141 m.
If we subsitute these “a” and “b” values into the Area of an Ellipse formula, the answer for the entire MCG including Grandstands is 59 800 square meters.
Area of MCG including Grandstands = 59 800 square meters.
Here are the “a” and “b” values for some other football stadium total sizes that we measured on Google Maps.
ANZ Stadium in Sydney = 130 m and 132 m
Gabba in Brisbane = 130 m by 95 m
Area of MCG Grandstands
Image Source: http://aftergrogblog.blogs.com
Area of Grandstands = Total Area – Area of Playing Oval
Area of Grandstands = 59 800 – 17 720
Area of MCG Grandstands = 42 080 square meters.
So how do 100 000 people fit into an Area of 42 080 square meters ?
Well the answer is that they would not comfortably fit into this area!
We need to remember that the Grandstands are multi-levelled, the levels also overlap each other, and the seating is on an incline.
This means the area of the seating in the MCG is actually a series of sections cut out from a giant cone.
Calculating this area would be a complex and lengthy task, and at the very least another whole separate blog post!
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