Tangled in Mathematics?

By Leanne Gunn

If you are yet to see Disney’s 2010 animated movie Tangled then you are really missing out! It is a good old fashioned fairy tale full of charming characters, frightful villains, catchy songs and frying pans…


Disney 2010

Loosely based on the German fairly tale Rapunzel, Tangled tells the tale of a princess kidnapped as a baby and locked away in a tall tower. Despite her ‘mothers’ tales of the horrors of the outside world, Rapunzel become curious and just before her 18th birthday she comes up with an elaborate ploy to send her ‘mother’ away and to persuade a thief called Flynn Rider to help her escape for a few days, just enough time to explore the outside world.

“And then I’ll brush, and brush, and brush, and brush my hair…”

Tangled Tower  CC-BY- Sam Howzit, via Flickr.

Tangled Tower
CC-BY- Sam Howzit, via Flickr.

A fair amount of Rapunzel’s charm is down to her long golden hair. Which she drops out of her tower window lifting her ‘mother’ up into her room.

This got me wondering… how long is Rapunzel’s hair?

The average human hair grows about 1.25 cm per month. From this we can work out how much, on average, hair grows a year (12 months):

1.25 cm x 12 months = 15 cm

That is not a lot! In fact 15 cm is the same length as a little ruler that you might carry around in a pencil case.

Now Rapunzel had plenty of time to grow her hair, she was trapped in that tower for 18 years. This means that on her 18th birthday her hair would have been:

15 cm x 18 years  = 270 cm long (or 2.70 m)

This might sound like a lot. However, lets compare this to something we are more familiar with. A standard UK door frame is just over 2 m high. In order for Rupunzel’s 2.7 m long hair to reach the floor when she dangled it out of her tower window, her tower could have only been as high as the upstairs in your house!

Record breaking hair

The Guinness World Record for the longest hair on a woman belongs to Xie Qiuping, China. She began growing her hair in 1973, age 13, and 31 years later, in 2004, it measured an extraordinary 5.63 m.

Using our estimates of hair growing at as rate of 15 cm per year we can see that Xie Qiuping’s hair grows faster than average, and is actually almost 1 m longer than we would have estimated.

In Tangled we see that her hair has some rather strange properties. As Rapunzel confesses to Finn, “I have magical hair that glows when I sing”, not many of us can say that. If her hair glows when she sings, perhaps it could also grow at a much faster rate than average!

“Rapunzel, Rapunzel, let down your hair..!”

Mother Gothel: Oh! Rapunzel! How you manage to do that every single day without fail! It looks absolutely exhausting, darling!

Rapunzel: Oh, it’s nothing.

Mother Gothel: Then I don’t know why it takes so long.

In Tangled, Rapunzel lifts up her mother to the top of the tower every day with just her hair. Is hair really strong enough to lift a fully grown adult?

Hair may be very thin, but it is also very strong. Hair is made from a strong protein called keratin, and a single strand can support up to 100 g. In the video below you’ll see that it has been calculated that a full head of hair could support the weight of two elephants!

Anastasia IV pulls the four-ton car (Gareth Fuller, The Sunday Times)

Anastasia IV pulls the four-ton car (Gareth Fuller, The Sunday Times)

Amazingly, there is a person who makes a living from the strength of her hair. Joanna Shawicka, also known as Anastasia IV, has hair so strong that she is the Guiness World Record holder for lifting the heaviest weight with her hair: pulling a 2.5 ton car 20 metres down a road. She uses two bottles of conditioner every day to keep her hair strong, and also has a team to help plait her hair in a certain way to help distribute the weight of objects evenly across her scalp. Any knots or tangles in her hair could affect its strength so it is brushed carefully each day.

Rapunzel’s hair is not plaited until later in the film when some children make a game of styling her hair at the lantern festival. In order to lift Mother Gothel up the tower she would have to make sure that her hair is tangle free, and that the weight of her mother is evenly spread in order to protect her head.

If Anastasia IV can lift a 2 ton car with her hair, it does seem possible that Rapunzel could lift her averagely weighted mother, especially if her tower is only 2.7 m high!

 Watch this video to find out more about what makes a human hair so strong!

For more Disney science, check out:

The Science of Disney

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Posted in Biology, Maths, Physics