E5.3 Circles, arcs and sectors

Learning objectives

E5.3.1Carry out calculations involving the circumference and area of a circle.• Carry out calculations involving the circumference and area of a circle.E5.3.2Carry out calculations involving arc length and sector area as• Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle. Answers may be asked for in terms of π. Formulas are given in the List of formulas. Includes minor and major sectors.

A circle is controlled by one length: its radius $r$ . The diameter crosses the centre from edge to edge, so $d=2r$ . Circumference measures the boundary and area measures the enclosed surface.

C=2\pi r=\pi d \qquad A=\pi r^2

First decide whether the given length is a radius or diameter. Convert units before substituting, keep $\pi$ on the calculator until the end, and attach linear units to circumference or square units to area. For an inverse problem, set the formula equal to the known value: divide by $2\pi$ to recover a radius from circumference, or divide by $\pi$ and then take the positive square root to recover a radius from area.

A circle has diameter $12$ cm. Then $r=6$ cm, so $C=12\pi$ cm and $A=36\pi\text{ cm}^2$ . If instead its area is $150\text{ cm}^2$ , then $r=\sqrt{150/\pi}=6.91\ldots$ cm. Use the requested accuracy; leave the answer in terms of $\pi$ when asked.

Do not square the diameter in $A=\pi r^2$ : halve it first. Circumference cannot be reported in square units, and area cannot be reported in linear units. Compound regions made from several circles belong to a later Topic.

A sector with central angle $\theta$ is the fraction $\theta/360$ of a full circle. The same fraction scales the full circumference to an arc length and the full area to a sector area.

Quantity	Full circle	Sector with angle $\theta$	Units
boundary along the curve	$2\pi r$	$\dfrac{\theta}{360}\times2\pi r$	length units
enclosed surface	$\pi r^2$	$\dfrac{\theta}{360}\times\pi r^2$	square units

Use the angle at the centre. For a minor sector with angle $\theta$ , use $\theta/360$ ; the matching major sector uses $(360-\theta)/360$ . Choose circumference for an arc or circle area for a sector, then multiply. If total sector perimeter is required, add the two radii to the arc length; those straight edges are not part of the arc.

For $r=9$ cm and minor angle $140^\circ$ , the minor arc length is $\frac{140}{360}(2\pi\times9)=7\pi$ cm. The minor sector area is $\frac{140}{360}(\pi\times9^2)=\frac{63\pi}{2}\text{ cm}^2$ . The major angle is $220^\circ$ , so the major sector area is $\frac{220}{360}(81\pi)=\frac{99\pi}{2}\text{ cm}^2$ .

Do not use an angle on the circumference as the sector angle, and do not scale the radius alone: scale the completed circumference or area formula. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc and belongs to a later Topic.

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—E5.4 Surface area and volumeRead

E5.3 Circles, arcs and sectors

Learning objectives

Calculate with circumference and circle area

Calculate arc length and sector area

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