In this blog, I’m going to use mathematics to calculate the arc length of Bristol’s most famous piece of engineering, The Clifton Suspension Bridge. All of the methods used appear in GCSE maths.
The Clifton Suspension Bridge opened in 1864 and was originally designed by Isambard Kingdom Brunel however sadly he was not able to see his designs come to fruition as he died a few years before the bridge opened.
What are we trying to calculate?
Below is a picture of a suspension bridge. I’ve turned the main arc of the bridge into the arc of a sector of a circle. We are trying to work out the length of the black arc. The sector has radius r and angle ɵ (this symbol is a Greek letter, theta, which is commonly used to denote an angle).
We’ll be using the sectors and arcs topic now.
Drawing this sector allows us to use a formula that we learn in higher tier GCSE maths – the formula for the length of an arc. This is a formula that is not given in the exam and so students need to remember it off-by-heart:
where r is the radius of the sector and ɵ is the angle – like in our diagram above.
So, all we need to do is find r and ɵ. And then substitute it into the formula and we’ve done it! It sounds easy, but let’s see just how easy it is…
Finding r
First of all, what do we know about the Clifton Suspension Bridge? We know that the length of the middle section is just over 214 metres (so we’re just going to round it to 214 metres) and the height of the towers (above the bridge floor) is 26 metres. Let me add these two measurements to the diagram from earlier:
Now, we can draw a line between the tops of the two towers to form a triangle. We know now that the base of the triangle is 214 metres.
Since the two shortest sides of the triangle are radii of a circle, they are the same. So the triangle is an isosceles triangle (a triangle with two equal lengths and angles).
Because it is isosceles, we can cut the triangle in half, like this, to form two identical right-angled triangles. Below is one of the triangles.
The angle and the base of the triangle have both been cut in half.
The hypotenuse of the triangle is r (the radius of the sector).
If you’re wondering how I know the height of the triangle, here’s how:
If you look back at our sector (before we turned it into a triangle), you can see that the height of the sector is a radius of the circle, therefore it is r.
We want to find the red line pictured below. Given that we know the height of the tower is 26 m (look at what I’ve circled in red), the red line must be 26 less than r.
Removing all the confusing shapes, this is what we have:
Going back to the right-angled triangle we created, let’s ignore the angle for now as we are trying to find r first.
We can use Pythagoras’ theorem here. Pythagoras’ theorem says that in a right-angled triangle a2 + b2 = c2 where c is the hypotenuse and a and b are the other two sides. This will form an equation that we can solve to find r.
If you’re not sure about Pythagoras’ theorem, read my Easy As Py guide and then try out my Pythagoras’ theorem worksheet.
If you’re unsure about some of the algebra involved, you can practice expanding brackets and solving equations by following the links.
Finding the angle
Now that we know r, we can find the angle, ɵ.
Let’s return to the right-angled triangle we had earlier:
We can substitute in our value of r from the previous step. I’m going to write 233 here to make things simpler. But remember we should use the full number (233.17…) when we do the next calculation. You should not do any rounding until the last step.
We can now use trigonometry (SOHCAHTOA) to find the angle.
I could actually use any of the three trig equations (as we know the length of all of the sides of the triangle). I’m going to choose SOH.
If you are unsure how SOHCAHTOA works, you can read my two blogs on how to find a missing side and, relevant to this example, how to find a missing angle.
You can also practice more trigonometry on our worksheet. You can also practice a mixture of trigonometry and Pythagoras questions, both of which we have used here, by trying our miscellaneous triangles worksheet.
Finding the arc length
Now we have the radius and the angle, we can finally answer the question.
For more practice of finding the length of an arc, try our sectors and arcs worksheet.
So there it is. The arc length of Bristol’s Clifton Suspension Bridge is 222 metres (to the nearest metre). That was a bit of a slog but I hope it was useful to see how some of the mathematics you learn at secondary school is applicable in the real world. Here we have used Pythagoras’ Theorem, SOHCAHTOA, a little bit of algebra and arc length.
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