Chapter 11 ยท Introduction

Chapter 11 โ€” Areas Related to Circles

A pizza slice, the sweep of a windscreen wiper, the beam of a lighthouse, the tip of a clock hand: each one is a fraction of a circle. In this chapter you learn to measure that fraction with a single idea โ€” compare the angle with a full turn.

A pizza of radius $14$ cm is cut into $8$ equal slices. How much crust does one slice carry, and how much cheese does it cover? You do not need a new formula: the whole crust is the circumference $2\pi r$, the whole top is the area $\pi r^2$, and one slice is exactly one eighth of each. The fraction is the same for the crust and for the cheese, because both come from the same angle at the centre: $\dfrac{45^\circ}{360^\circ}=\dfrac18$.

That single idea โ€” a part of a circle is the fraction $\dfrac{\theta}{360^\circ}$ of the whole โ€” gives every formula of this chapter. Section 11.1 names the part cut off by two radii (a sector) and finds its arc length and area; it also handles the hands of a clock and the sweep of a wiper. Section 11.2 looks at the part cut off by a chord (a segment) and shows that it is a sector with a triangle removed. In this course the segments have central angles of $60^\circ$, $90^\circ$ or $120^\circ$, where the triangle is easy to handle.

Circle: $2\pi r$ and $\pi r^2$โ†’Central angle $\theta$ as a fraction $\dfrac{\theta}{360^\circ}$โ†’Arc length and sector areaโ†’Segment = sector โˆ’ triangleโ†’Perimeters and areas of designs
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • define a sector, an arc and the central angle, and tell a minor sector from a major one;
  • find the length of an arc and the area of a sector from the radius and the angle, and find the radius or the angle when the other quantities are given;
  • find the perimeter of a sector and the area of a ring-shaped sector between two concentric arcs;
  • find the area swept by a clock hand, a wiper or a tethered animal's rope;
  • find the area of a minor segment and a major segment when the central angle is $60^\circ$, $90^\circ$ or $120^\circ$;
  • state the value of $\pi$ you use, give answers to two decimal places, and attach the correct unit (cm, cmยฒ, m, mยฒ).
Teaser โ€” Cutting a Pizza into Equal Slices

A pizza of radius $14$ cm (we use $\pi=\tfrac{22}{7}$) is cut into $n$ equal slices. Change $n$ with the slider, or press Play. The angle of one slice is $\dfrac{360^\circ}{n}$; its crust and its cheese are the same fraction $\dfrac1n$ of the whole crust ($88$ cm) and the whole top ($616$ cmยฒ).

8
angle of a sliceโ€”
fraction of the pizzaโ€”
crust of one slice (cm)โ€”
area of one slice (cmยฒ)โ€”

Sections in this chapter

11.1

Sectors: Arc Length and Area

The sector and its central angle, the length of an arc, the area of a sector, and problems on clock hands, wipers and ropes.

$l=\dfrac{\theta}{360^\circ}\,2\pi r$, $\ A=\dfrac{\theta}{360^\circ}\,\pi r^2$
Open section โ†’11.2

Segments of a Circle

Minor and major segments: the area of a segment as sector minus triangle, for central angles of $60^\circ$, $90^\circ$ and $120^\circ$.

$A_{\text{seg}}=A_{\text{sector}}-A_{\triangle}$
Open section โ†’
โœ… Before you start
You need the circumference $2\pi r$ and the area $\pi r^2$ of a circle (Class 9 and earlier), the radius and diameter, and the angle sum of a triangle. The area of an equilateral triangle of side $a$ is $\dfrac{\sqrt3}{4}a^2$ and the area of a triangle is $\dfrac12\times\text{base}\times\text{height}$. For the segments in Section 11.2 it helps to know the values $\sin60^\circ=\dfrac{\sqrt3}{2}$ and $\cos60^\circ=\dfrac12$ from Section 8.2, and the facts about circles from Chapter 10 (radius, chord, the radius to a point on a circle). Warm up: a circle has radius $7$ cm. With $\pi=\tfrac{22}{7}$, its circumference is $44$ cm and its area is $154$ cmยฒ. What fraction of the circle is a sector of angle $90^\circ$? (Answer: $\tfrac14$.)
โ† 10.2 Tangents from a Point Outside a Circle11.1 Sectors: Arc Length and Area โ†’