Ben is sporting combats as he prepares to plunge down an aerial ropeway for the sake of investigating Pythagoras’ famous theorem. He hopes the army have understood how to apply the theorem correctly, or his stunt could go horribly wrong!

00.41—02.19

Katie reviews the key features of right-angled triangles and looks in detail at the special relationship between the sides that Pythagoras expressed in his theorem. We see how the areas of the squares on the shorter sides fit exactly into the area of the square on the hypotenuse, and how to express this algebraically.

02.19—03.13

Ben starts with a different right-angled triangle and proves, using areas, that the formula c² = a² + b² still works. (The demonstration is based on ‘Perigal’s Dissection’.)

03.13—05.23

Returning to the army training camp, Ben looks again at the slide and shows how the length of the rope can be calculated because it forms the hypotenuse of a right-angled triangle.

05.23—06.19

Ben does the calculation and introduces the idea of taking a square root. He discusses how to find the square-root function on a calculator and how to round an answer sensibly. With the dimensions of the slide established, Ben glides down the hypotenuse — and just about manages to keep his cool as he lands!

06.19—09.01

Katie reveals that many civilisations, including the Chinese, Babylonian and Egyptian, knew of and used the relationship whose ‘discovery’ is ascribed to Pythagoras. She tells us about the work of Egyptian surveyors, who were known as ‘rope-stretchers’ — and hopes for a trip to the Pyramids!

She finds herself not in Egypt but at Bradford City football ground to check out corners with the help of the youth team. Using some knotted ropes and some well-known Pythagorean triples, Katie discovers a practical method for making sure the corners are right-angled.

09.01—11.38

In ‘Tick or Trash’, Ben and Katie tackle another problem involving rope lengths. They reiterate the steps used in applying Pythagoras’ Theorem — but Ben makes a common mistake in his calculation. He then explains how he should have spotted it.

11.38—13.14

On a windswept site, Ben braves the elements to discover how Pythagoras’ Theorem is used by a team of engineers who are setting up a wind farm. They use it to establish exactly where to construct anchor points for steel guy ropes so that they remain taut.

13.14—end

We see how to rearrange the equation c² = a² + b² to find the length of one of the shorter sides of a right-angled triangle. Having completed most of the maths, despite the wind and snow, Ben invites viewers to solve the problem of where to fix the ropes.