Geometry Concepts

1. Pythagoras Theorem, Sine Rule, and Cosine Rule

1.1 Pythagoras Theorem

Definition

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

c^2 = a^2 + b^2

Where:

$c$ : Hypotenuse
$a, b$ : Other two sides

Example

A triangle has sides $3 \, \text{cm}$ and $4 \, \text{cm}$ , and the hypotenuse is unknown.

c^2 = 3^2 + 4^2 = 9 + 16 = 25

c = \sqrt{25} = 5 \, \text{cm}

Applications

Navigation
Architecture
Distance measurement

1.2 Sine Rule

Formula

For any triangle:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Where:

$a, b, c$ : Lengths of the sides
$A, B, C$ : Opposite angles

Example

Given $A = 30^\circ, B = 60^\circ, a = 10 \, \text{cm}$ , find $b$ .

\frac{a}{\sin A} = \frac{b}{\sin B}

\frac{10}{\sin 30^\circ} = \frac{b}{\sin 60^\circ}

\frac{10}{0.5} = \frac{b}{0.866}

b = 17.32 \, \text{cm}

1.3 Cosine Rule

Formula

For any triangle:

c^2 = a^2 + b^2 - 2ab \cos C

Where:

$a, b, c$ : Sides of the triangle
$C$ : Angle opposite side $c$

Example

Given $a = 7 \, \text{cm}, b = 10 \, \text{cm}, C = 60^\circ$ , find $c$ :

c^2 = 7^2 + 10^2 - 2(7)(10)\cos 60^\circ

c^2 = 49 + 100 - 70

c^2 = 79, \, c = \sqrt{79} \approx 8.89 \, \text{cm}

Applications

Surveying
Engineering designs
Astronomy

2. Lengths of Arcs, Perimeters of Sectors, and Segments

2.1 Length of an Arc

Formula

\text{Arc Length} = \theta \times r

Where:

$\theta$ : Angle in radians
$r$ : Radius of the circle

Example

A circle has a radius of $7 \, \text{cm}$ , and $\theta = \frac{\pi}{3}$ :

\text{Arc Length} = \frac{\pi}{3} \times 7 = \frac{7\pi}{3} \approx 7.33 \, \text{cm}

2.2 Perimeter of a Sector

Formula

\text{Perimeter} = 2r + \text{Arc Length}

Example

Using the previous arc length ( $7.33 \, \text{cm}$ ) and $r = 7 \, \text{cm}$ :

\text{Perimeter} = 2(7) + 7.33 = 14 + 7.33 = 21.33 \, \text{cm}

2.3 Segment of a Circle

A segment is the area between a chord and an arc.

Area of Segment:

\text{Area} = \frac{1}{2}r^2(\theta - \sin \theta)

Example

For a radius of $7 \, \text{cm}$ and $\theta = \frac{\pi}{3}$ :

\text{Area} = \frac{1}{2}(7)^2\left(\frac{\pi}{3} - \sin\frac{\pi}{3}\right)

3. Longitudes and Latitudes

3.1 Definition

Longitude: Angular distance east or west of the Prime Meridian, measured in degrees ( $0^\circ$ to $180^\circ$ ).
Latitude: Angular distance north or south of the Equator, measured in degrees ( $0^\circ$ to $90^\circ$ ).

3.2 Distance Between Two Points

Formula

\text{Distance} = R \times \Delta\theta

Where:

$R$ : Radius of the Earth ( $\approx 6371 \, \text{km}$ )
$\Delta\theta$ : Angular difference in radians

Example

Two points have a latitude difference of $30^\circ$ :

\Delta\theta = \frac{30 \times \pi}{180} = \frac{\pi}{6}

\text{Distance} = 6371 \times \frac{\pi}{6} \approx 3336 \, \text{km}

3.3 Real-World Applications

Navigation: Calculating shortest flight routes.
Geography: Measuring distance between cities.
Astronomy: Locating celestial bodies.

3.4 Common Misconceptions

Confusing latitude (horizontal) and longitude (vertical).
Assuming Earth's radius is constant (it varies slightly due to its oblate shape).

4. Summary

Pythagoras Theorem, Sine Rule, and Cosine Rule are essential for solving triangles and measuring distances.
Lengths, perimeters, and areas of circles, sectors, and segments are calculated using radians and circle formulas.
Longitude and Latitude aid in understanding Earth’s spherical geometry and calculating distances between global points.

These concepts are foundational in navigation, astronomy, engineering, and practical problem-solving in everyday life.