CAIE A-Level Mathematics 1.4 Circular Measure Question Bank

Practise working in radians and combining arc length, sector area, chord and triangle geometry to solve perimeters, segments, shaded regions and changing-area problems.

[Maximum number: 7]

Figure for Question 1.4 Circular measure question 1 — CAIE A-Level Mathematics AS

The diagram shows a motif formed by the major arc A B of a circle with radius r and centre O, and the minor $\operatorname{arc} A O B$ of a circle, also with radius r but with centre C. The point C lies on the circle with centre O.

(a)

Given that angle $A C B=k \pi$ radians, state the value of the fraction k.

[ 1 ]

$k=\frac{2}{3}$

B1

Marking guidance:

Allow $\mathrm{ACB}=\frac{2 \pi}{3}$ .

(b)

State the perimeter of the shaded motif in terms of $\pi$ and r.

[ 1 ]

Perimeter of shaded area $=2 \pi r$

B1

(c)

Find the area of the shaded motif, giving your answer in terms of $\pi, r$ and $\sqrt{3}$ .

[ 5 ]

Major sector $O A B=\frac{1}{2} r^{2} \times \frac{4 \pi}{3}$

*M1

Expect $\frac{2}{3} \pi r^{2}$ . Finds area of any relevant sector or
triangle. Can be embedded in segment formula.

One or both segments $=[2] \times\left(\frac{1}{2} r^{2} \times \frac{\pi}{3}-\frac{1}{2} r^{2} \sin \frac{\pi}{3}\right)$

*M1

$=[2]\left(r^{2} \frac{\pi}{6}-r^{2} \frac{\sqrt{3}}{4}\right)$

A1

Shaded area $=\frac{2}{3} \pi r^{2}-2\left(\frac{1}{6} \pi r^{2}-\frac{r^{2} \sqrt{3}}{4}\right)$

DM1

$=\frac{\pi r^{2}}{3}+\frac{r^{2} \sqrt{3}}{2}$

A1

6(c)

Alternative method for Question 6(c)

Sector CAOB $=[2] \times \frac{1}{2} r^{2}$ their $\frac{1}{3} \pi$

*M1

Expect [2] $\times \frac{1}{6} \pi r^{2}$ .
Can be embedded in segment formula.

One or both segments $=[2] \times\left(\frac{1}{2} r^{2} \times \frac{\pi}{3}-\frac{1}{2} r^{2} \sin \frac{\pi}{3}\right)$

*M1

$=[2]\left(r^{2} \frac{\pi}{6}-r^{2} \frac{\sqrt{3}}{4}\right)$

A1

Shaded area $=\pi r^{2}-\left\{\frac{1}{3} \pi r^{2}+2\left(r^{2} \frac{\pi}{6}-r^{2} \frac{\sqrt{3}}{4}\right)\right\}$

DM1

$=\frac{\pi r^{2}}{3}+\frac{r^{2} \sqrt{3}}{2}$

A1

6(c)

Alternative method for Question 6(c)

Area of rhombus AOBC $=[2] \times \frac{1}{2} r^{2} \sin \frac{\pi}{3}$

M1

Expect [2] $\times \frac{\sqrt{3}}{4}$ .
Can be embedded in segment formula.

One or both segments $=[2] \times\left(\frac{1}{2} r^{2} \times \frac{\pi}{3}-\frac{1}{2} r^{2} \sin \frac{\pi}{3}\right)$

M1

$=[2]\left(r^{2} \frac{\pi}{6}-r^{2} \frac{\sqrt{3}}{4}\right)$

A1

Shaded area $=\pi r^{2}-\left\{\frac{\sqrt{3}}{2} r^{2}-4\left(r^{2} \frac{\pi}{6}-r^{2} \frac{\sqrt{3}}{4}\right)\right\}$

DM1

$=\frac{\pi r^{2}}{3}+\frac{r^{2} \sqrt{3}}{2}$

A1

[Maximum number: 2]

Figure for Question 1.4 Circular measure question 2 — CAIE A-Level Mathematics AS

The diagram shows a sector of a circle with centre O and radius $r \mathrm{~cm}$ . The shaded region is bounded by the chord AB and the $\operatorname{arc} A B$ . The size of angle AOB is $\frac{2}{3} \pi$ radians.

Show that the area of the shaded region is approximately $0.614 r^{2} \mathrm{~cm}^{2}$ .

It is given that the radius of the circle is increasing at a rate of $0.4 \mathrm{~cm} \mathrm{~s}^{-1}$ .

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CAIE A-Level Mathematics 1.4 Circular Measure Question Bank

Exam points

Covered syllabus points

Covered syllabus points

1.4 Circular measure question 1

Question (a)

Question (b)

Question (c)

1.4 Circular measure question 2