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Triangles and TrigonometryIntroduction

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By the early 19th century, explorers had discovered most of the world. Trade and transportation was booming between distant countries, and this created a need for accurate maps of the entire planet.

Today we have satellites that can take photos from above – but 200 years ago, creating maps was a difficult and time consuming task. It was done by mathematicians like Radhanath Sikdar, who worked on the Great Trigonometrical Survey: a century-long project to measure all of India, including the Himalayan mountain range.

The theodolite, a surveying instrument

Of particular interest was the quest to find the highest mountain on Earth. There were a few different candidates, but from hundreds of kilometers away, it was difficult to tell which one was the highest.

So how do you measure the height of a mountain?

Artboard 1aa

Today we can use satellites to measure the height of mountains to within a few centimeters – but these did not exist when Radhanath was surveying India.

Climbers use altimeters to determine their altitude. These devices use the difference in air pressure at different heights. However this would have required someone to actually climb to the top of every mountain – an extremely difficult feat that was not achieved until a century later.

You could also try using similar triangles, like we did in the previous course. This method requires knowing the distance to the base of the mountain: the point at sea level that directly below its summit. We can do this for trees or tall buildings, but for mountains this point is hidden underneath hundreds of meters of rock.

Edmund Hillary and Tenzing Norgay were the first to reach the top of Mount Everest, in 1953.

But there are more advanced geometric techniques, which Radhanath used to discover the highest mountain on Earth: it is now called Mount Everest. His measurement is only a few meters off today’s official height of 8848 meters.

In this course you will learn about many different features and properties of triangles. These will allow you to measure the height of mountains, but they are also of fundamental importance in many other areas of mathematics, science and engineering.

Triangles are special because they are particularly strong. They are the only polygon that, when made out of wooden beams and hinges, will be completely rigid – unlike rectangles, for example, which you can easily push over.

COMING SOON – Animations

This property makes triangles particularly useful in construction, where they can carry heavy weights.

A “Truss bridge” is supported by triangular bars

Triangles in high-voltage electricity Pylons

Even bikes use triangles for stability.

Triangles are also the simplest polygon, with the fewest sides. This makes them particularly well suited to approximating complex curved surfaces. This is done in physical building…

“The Gherkin”, a skyscraper in London

Bank of China Tower in Hong Kong

Courtyard of the British Museum in London

…but also in virtual worlds. In computer generated graphics (e.g. for movies or video games), all surfaces are approximated using a “mesh” of tiny triangles. Artists and software engineers need to know about geometry and trigonometry in order to be able to move and transform these triangles realistically, and to calculate their colour and texture.