Trigonometry · Lesson 5 (synthesis) · Pearson Edexcel International A-Level Pure Mathematics 1
This is a synthesis lesson. You already have every tool you need from Lesson 4 — radians, arc length s = rθ, and sector area ½r²θ. Here we put them to work on richer problems, and we meet one new shape that falls out almost for free: the segment.
First, the toolkit you are bringing into this lesson — all with θ in radians:
| Quantity | Formula | In words |
|---|---|---|
| Arc length | s = rθ | radius times angle |
| Sector area | A = ½r²θ | the area of the whole slice |
Now the new shape. Slice a circle and you get a sector (the whole wedge). Join the two ends of the arc with a straight line — a chord — and the sector splits into two pieces: a triangle (the two radii and the chord) and a segment (the curved sliver between the chord and the arc).
So if we can find the sector and the triangle, the segment is just the difference. Watch the formula get built, one step at a time.
So, with θ in radians, the synthesis toolkit for this lesson is:
| Quantity | Formula |
|---|---|
| Triangle (two radii, included angle) | ½r² sin θ |
| Segment (sector − triangle) | ½r²(θ − sin θ) |
| Chord length | 2r sin(θ/2) |
| Segment perimeter (arc + chord) | rθ + 2r sin(θ/2) |
Here is something that surprises most people: trigonometry did not begin with the sine. It began with the chord — the very line you just drew across the sector.
Around the second century BCE, the Greek astronomer Hipparchus of Nicaea wanted to predict the positions of the Sun, Moon, and stars. To do that he needed to turn angles into lengths, so he built what is generally regarded as the first trigonometric table — a table of chords: for each central angle, how long is the chord that spans it? Three centuries later, Ptolemy refined this into the famous chord table in his Almagest, computed in steps of half a degree on a circle of radius 60.
Their chord function, written crd θ, is exactly the formula in your toolkit. For a circle of radius r, the chord of an angle θ is 2r sin(θ/2) — so the ancient crd and our modern sine are the same idea wearing different clothes. The sine we use today emerged later, refined by mathematicians in India (such as Āryabhaṭa) and the medieval Islamic world, as a more convenient half-chord. When you compute a chord in this lesson, you are using the oldest tool in trigonometry.
A curiosity for the curious: long before all this, Hippocrates of Chios (5th century BCE) studied crescent shapes bounded by circular arcs — close cousins of the segment — and managed to show that the area of a particular "lune" exactly equals the area of a triangle. It was one of the first times anyone turned a region bounded by curves into a region bounded by straight lines, an early step in the long quest to "square the circle."
Make the idea your own. Pick whichever of these helps you think — write, sketch, or say it out loud:
There is no single right wording here. The goal is to be able to say what a segment is and why its formula looks the way it does — because an idea you can explain is one you can use under pressure.
Here is a real engineering problem that comes down entirely to a segment. You will solve it the way an engineer would — one decision at a time. Each step unlocks only when the previous one is right, and the chart fills in as you go, so you can always see what you know.
The situation. A water tank is a cylinder lying on its side, of radius r = 1.2 m and length 3 m. The water is 0.45 m deep at the lowest point. Slice the tank straight across and you see a circle: the flat water surface is a chord, and the wet area is the segment beneath it. The question: how much water (by volume) is in the tank?
Drop a perpendicular from the centre O to the water surface. Its length is r − h = 1.2 − 0.45 = 0.75 m, and it splits the angle into two halves, so cos(θ/2) = 0.75 / 1.2.
Find θ in radians (2 d.p.):
Use ½r²θ with your value of θ.
Sector area in m² (2 d.p.):
The two radii and the chord make an isosceles triangle: ½r² sin θ.
Triangle area in m² (2 d.p.):
Segment = sector − triangle.
Wet cross-section in m² (2 d.p.):
The cross-section is the same all along: volume = wet area × length (3 m).
Volume in m³ (2 d.p.):
Tip: keep full accuracy on your calculator between steps and round only the figure you type. A real tank of these dimensions holds a little under two cubic metres — about two thousand litres — when filled to this depth.
Notice the shape of the work: the hard part was not any single formula, it was seeing that the wet area is a segment, and finding θ from the depth. Once those two moves are made, the rest is the toolkit you already had.
Radians, arcs, and sectors are not only on the page — they are in how your body moves. Hold your arm straight and sweep your hand in an arc: your arm is the radius r, the angle your shoulder turns through is θ, and your fingertip travels an arc of length s = rθ. The area your arm can sweep is a sector; the patch just beyond your reach, cut off by a straight edge in front of you, is a segment.
Take a moment to notice how you felt working through the tank problem:
Design and solve a segment problem of your own. Invent a real situation whose answer depends on the area of a segment — for example a partly-filled pipe or trough, an arched window or doorway, a tunnel cross-section, or a slice cut from a round cake. Then write a full model solution that:
If you can, swap with a partner and solve each other's problems — that is the real test of whether your setup is clear.
How your work will be assessed — the Haven Maths Rubric:
| Strand | What we are looking for here |
|---|---|
| Conceptual Understanding | You can explain that a segment is a sector minus its triangle, and why θ must be in radians. |
| Fluency & Accuracy | Correct substitution into ½r²θ, ½r² sin θ, and ½r²(θ − sin θ); full accuracy carried through; sensible final rounding. |
| Application to Problems | You translate an unstructured, real-world situation into a labelled circle, recovering θ from the given information. |
| Independence & Reflection | You design a coherent problem of your own and check that your answer is reasonable. |
Ready to build fluency? The practice companion has exam-style questions on arcs, sectors, and segments, with Socratic support whenever you get stuck.
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