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# ACMGM018

## 2.8 Shortest Distance between Points on a Parallel

Note: if you cannot remember what parallels of longitude are, revise notes for 2.6 Modelling the World.

Note: if you cannot remember how to find the shortest distance between points on great and small circles, revise notes for 2.5 Distance Between Points on a Sphere.

### Shortest Distance between Points on the Equator

• Locations of equal latitude exist on the same parallel of longitude. By taking the shorter section of that parallel with the locations as the endpoints, we can extract an arc representing the shortest path between the two locations. The length of that path can be found using the longitudinal coordinates of the locations.
• The plane of the equator intersects the centre of the world and so it is a great circle.
• We can modify the equation for arc length (from notes 2.1 Circles and Arcs) for this situation. There are 3 situations with different formulas:
• If both locations are given in the same units (oE or oW) or one or both lie on the Greenwich meridian, the formula is:
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## 2.2 Area of Sectors and Segments

Note: if you cannot remember how to calculate values of a circle, review notes for 2.1 Circles and Arcs.

### Area of a Sector

• The area of a circle sector can be found by multiplying the formula for the area of a regular circle by the fraction of the circle which the sector encompasses. The angle is used for this fraction:

A=\frac{\pi r^{2} \theta}{360}

Where A is the area of the sector, r is the radius of the sector and θ is the angle of the sector.

Example

Above is a diagram of a machine component which is manufactured by cutting a 45o sector out of circle of radius 15mm. The company making these parts wishes to know the area of material which is wasted (cut out) per component. To do this we will use our area formula:

A=\frac{\pi r^{2} \theta}{360}=\frac{\pi 15^{2} 45}{360}=88.36 m m^{2}

### Chord

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## 2.1 Circles and Arcs

### Circle

• The set of all points in the plane that are a fixed distance (the radius, r) from a fixed point (the centre, O).
• The angle at the centre of a circle is 360°.

### Diameter (D)

• A chord passing through the centre.
• D = 2r, twice the length of the radius.

### Circumference of a Circle

• The circumference of a circle, AKA the perimeter, is given by:

c=2 \pi r

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