A-level Mathematics/OCR/C2/Circles and Angles

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Angular Measurement and Circular Sectors[edit]

A section of a circle is known as a sector. One side of the sector is the radius. The portion of the cirumference that is included in the sector is known as an arc.


Angular Degree[edit]

A circle has 360 degrees. Each degree can have 60 minutes (designated as ' ) and each minute can have 60 seconds (designated as " ). Since we can not convert minutes or seconds into radians directly we need to convert the minutes and seconds into a decimal number. Here is the formula:

Convert X^\circ Y' Z" into degrees.

X + \left (Y \times \frac{1}{60} \right ) + \left (Z \times \frac{1}{3600} \right ) = X + \frac{Y}{60}+ \frac{Z}{3600}

For a practical example, convert 23^\circ 18' 38" into degrees.

23 + \frac{18}{60}+ \frac{38}{3600} = 23.3106

Angular Radian[edit]

A radian is the angle subtended at the centre of a circle by an arc of its circumference equal in length to the radius of the circle. Since we know that that the formula for the circumference of a circle is C = 2 \pi r we can determine that there are 2 \pi radians in a circle. We abbreviate radians as rad.

Conversion Between Degrees and Radians[edit]

Mathematics requires us to use radians for most angular measurements. Therefore we need to know how to convert from degrees to radians Since we know that there are 360^\circ degrees or 2 \pi radians in a circle. We can determine these equations:

1^\circ = \frac {\pi }{180}

1\ radian = \frac {180}{\pi }

So we can write these general formulae.

X^\circ = \frac {X \times \pi }{180}

X\ radian \left(s \right ) = \frac {X \times 180}{\pi }

Here is an example convert 20^\circ into radians

20^\circ = \frac {20 \times \pi }{180} = \frac{1}{9} \pi\ rad

Convert \frac{1}{9} \pi into degrees.

\frac{1}{9} \pi\ radian \left(s \right ) = \frac {\frac{1}{9} \pi \times 180}{\pi } = 20^\circ

Arc Length[edit]

In most cases it is very difficult to measure the length of an arc with a ruler. Therefore we need to use a formula in order to determine the length of the arc. The formula that we use is:

Arc\ Length = ( \theta\ in\ radians)(radius)\, in symbols this is s= \theta r\,. Note: θ need to be in radians

Here is an example, determine the length of the arc created by a sector with a 6cm radius and an angle of 53^\circ.

The first thing we need to do is convert θ from degrees to radians.

53^\circ = \frac {53 \pi}{180}rad

Now we can calcuate the length of the arc.

s = \frac {53 \pi}{180} \times 6 \approx 5.55cm

Area of a Sector[edit]

The area of a sector can be found using this formula:

Area = \frac{1}{2}r^2 \thetaNote: θ need to be in radians.

Calculate the area of a sector with a 3cm radius and an angle of 2π.

Area = \frac{1}{2}3^2 \times 2 \pi \approx 28.3cm^2

This is part of the C2 (Core Mathematics 2) module of the A-level Mathematics text.

Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration

Appendix A: Formulae