A sector of radius 6 and angle π/3 has area 18.85. The segment inside it has area 3.26, because the triangle takes up 15.59 of it. Give the sector where the segment was wanted and you are not close.
Three regions, one diagram. The words sector, triangle and segment name different areas, and the numbers are not close.
One radian is the angle whose arc is one radius long. Walk round the circle and 2π radii fit, so a full turn is 2π radians, and
π radians = 180°, 1 radian = 57.3°, 60° = 1.047 radians
Then the two formulas have no fraction in them at all:
With r = 6 and θ = π/3: the arc is 6 × π/3 = 2π = 6.28, and the sector is ½ × 36 × π/3 = 6π = 18.85.
On Paper 1 write 6π. Analysis Paper 1 has no calculator, the question will be built so the answer is exact, and 18.85 is not an exact answer. The guidance says radian measure may be given as an exact multiple of π or as a decimal, and which one is wanted is decided by whether you have a machine.
Both formulas need radians. Put 60 into ½r²θ instead of π/3 and you get 1080, when the entire circle has area 113.1. The check is free: a sector is part of its circle, so its area must be under πr² and its arc under 2πr. That catches every error big enough to matter.
| Region | Bounded by | Area, r = 6, θ = π/3 |
|---|---|---|
| Sector | Two radii and the arc | ½r²θ = 6π = 18.85 |
| Triangle | Two radii and the chord | ½r²sinθ = 9√3 = 15.59 |
| Segment | The chord and the arc | sector − triangle = 3.26 |
The two area formulas look almost identical and are not: ½r²θ uses the angle, ½r²sinθ uses its sine. At θ = π/3 that is 1.047 against 0.866, so the sector is the larger, which it has to be: the triangle sits inside it.
A segment is a sector with the triangle taken out.
18.85 − 15.59 = 3.26
and on this sector the triangle is 83% of the whole thing, which is why the answers are so far apart. Read the question for the word. "The area between the chord and the arc" is a segment; "the area enclosed by the two radii" is a sector. A diagram with the region shaded is doing you a favour, so look at it.
Perimeter is not area, and a sector has two straight edges. The perimeter of this sector is the arc plus both radii:
6.28 + 6 + 6 = 18.28
Giving 6.28 is the commonest answer, and it is the arc on its own. Trace the shape with a finger: you go along a radius, round the arc, and back along the other radius.
Both formulas rearrange, and the angle is usually what is missing. A sector of radius 5 has area 30. Then
30 = ½ × 25 × θ, so θ = 60/25 = 2.4 radians
which is 137.5°, under the 2π of a full turn, so it is a real sector. Always check the angle you get is below 2π. If it is not, you have probably used the arc formula on an area, or the other way round.
Neither formula contains a trigonometric function, so the angle MODE does not affect the arithmetic here. What the mode affects is the triangle's ½r²sinθ, which is where a segment question goes wrong.
When you may use it. Analysis Paper 1 is non-calculator, and that is where 6π and 9√3 are the answers. On Paper 2 the decimals are fine, and the machine's only real job is the sine in the triangle.
The mark people lose. Giving the sector when the question said segment. 18.85 against 3.26 is not a rounding, and no sanity check on size will save you, because both are comfortably inside the circle. The habit: before any arithmetic, shade the region on the diagram with your pencil. If your shading has a straight chord across it, you need the triangle subtracted; if it has two straight radii, you do not.
Throughout: a circle of radius 6, with a sector of angle π/3.
1. Find the sector's area, to 2 decimal places.
2. Find the triangle's area, to 2 decimal places.
3. So find the segment's area, to 2 decimal places.
4. Find the sector's perimeter, to 2 decimal places.
5. Why is ½r²θ bigger than ½r²sinθ here?
1 markThe correct formula for the region asked for.
1 markThe angle in radians.
1 markThe subtraction, on a segment.
1 markThe answer, exact on Paper 1 and with units.
On a segment question the subtraction is its own mark, so a student who finds both areas correctly and forgets to subtract scores two of four rather than nothing. Write both areas down, then subtract on a new line.
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