This module covers the use of measurement, geometry and trigonometry to formulate and solve problems involving angle, length, area and volume in two and three dimensions, with respect to objects, the plane and the surface of the earth.

Measurement and trigonometry, including:

- calculation of surface area and volume of spheres, cylinders, cones, pyramids and prisms, and their composites
- application of linear scale factor \( k > 0 \) of similar figures and shapes to scale lengths, areas and volumes with practical applications
- review of the methods for solving right and non-right-angled triangles, including the ambiguous case of the sine rule, and their application to solving practical problems in two and three dimensions
- specification of location (distance and direction) in two dimensions using three-figure bearings with applications such as navigation and orienteering, including situations involving the solution of non-right-angled triangles.

Spherical geometry, including:

- circle mensuration; arc length using the rule \( \displaystyle s = r \times \frac{\pi}{180} \times \theta^{\circ} \) with practical applications
- arc length of a sector of a circle, and the areas of sectors and segments with practical applications
- use of trigonometry and Pythagoras’ theorem in two and three dimensions to solve problems involving the solution of right-angled triangles within a sphere
- use of a sphere of radius 6400 km as a model of the earth, and meridians and parallels and their use in locating points on the surface of the earth in terms of latitude and longitude (specified in decimal degrees) using the Greenwich meridian and the equator as reference
- use of meridians to determine the shortest distance from any point on the earth to a pole or the equator
- use of a great circle to determine the shortest distance between two points on the surface of the earth that have the same longitude
- use of 15° of longitude as equating to a 1 hour time difference to identify time zones, and determine travel times of journeys that cross two or more time zones from departure and arrival times.

*source – VCE Mathematics Study Design*

Lessons

#### Surface Area of Prisms

- Video - Surface Area of Rectangular Prisms (4:51)
- Video - Surface Area of Triangular Prisms (2:10)
- Topic - Total Surface Area of Open Rectangular Prisms
- Topic - Total Surface Area of Rectangular Prisms
- Topic - Side of Rectangular Prisms
- Topic - Total Surface Area of Triangular Prisms using Nets 1
- Topic - Total Surface Area of Triangular Prisms using Nets 2

#### Surface Area of Pyramids

- Video - Surface Area of Square Pyramids (4:56)
- Topic - Total Surface Area of a Square Pyramid given Base Side and Slant Height
- Topic - Total Surface Area of a Square Pyramid given Base Side and Height
- Topic - Total Surface Area of a Square Pyramid given Base Side and Side Height
- Topic - Total Surface Area of a Square Pyramid given Hieght and Slant Height
- Topic - Finding Slant Height by Total Surface Area of a Square Pyramid

#### Proof of Similar Triangles

- Video - Proofs of Similar Inscribed Triangles (8:10)
- Video - Proofs of Similar Triangles using Parallel Lines (4:28)
- Video - Proving Trapeziums by Proofs of Similar Triangles using Ratios (7:06)
- Video - Proofs of Similar Triangles using Ratios & Vertically Opposite Angles (3:33)
- Video - Proofs of Similar Triangles using Midpoints & Parallel Lines (11:17)
- Video - Proofs of Similar Triangles using Ratios & Angles (4:25)

#### Angles of Elevation and Angle of Depression

- Video - Angles of Elevation 1 (4:07)
- Video - Angles of Elevation 2 (2:48)
- Video - Angles of Elevation 3 (4:44)
- Topic - Angles of Elevation by Sine Ratio
- Topic - Angles of Elevation by Cosine Ratio
- Topic - Angles of Elevation by Tangent Ratio
- Topic - Angles of Depression by Sine Ratio 1
- Topic - Angles of Depression by Sine Ratio 2
- Topic - Angles of Depression by Sine Ratio 3
- Topic - Angles of Depression by Sine Ratio 4
- Topic - Angles of Depression by Cosine Ratio 1
- Topic - Angles of Depression by Cosine Ratio 2
- Topic - Angles of Depression by Cosine Ratio 3
- Topic - Angles of Depression by Cosine Ratio 4
- Topic - Angles of Depression by Tangent Ratio 1
- Topic - Angles of Depression by Tangent Ratio 2
- Topic - Angles of Depression by Tangent Ratio 3
- Topic - Angles of Depression by Tangent Ratio 4

#### True Bearings

- Video - State the True Bearing (5:30)
- Video - Sketching True Bearing (1:54)
- Video - Finding True Bearing (4:27)
- Video - Distance using True Bearing (6:28)
- Video - Understanding of Bearing (4:29)
- Video - Sketching Diagrams using Bearing (3:33)
- Video - Finding the Bearing (8:56)
- Video - Finding the Distances using Bearing (4:49)

#### Sine Rule

- Video - Finding Sides using Sine Rule (6:42)
- Video - Finding Angles using Sine Rule (4:24)
- Video - Sine Rule for Finding Sides (7:02)
- Video - Sine Rule for Finding Angles (6:06)
- Video - Sine Rule with Obtuse Angles (3:47)
- Video - Sine Rule Formula (2:49)
- Video - Finding Unknown Sides using Sine Rule (6:53)

#### Applications of Sine Rule

- Video - Sine Rule in Triangular Fields (5:27)
- Video - Sine Rule involving Acute and Obtuse Angle Bearings (5:45)
- Video - Sine Rule in Isosceles Triangles (2:29)
- Video - Sine Rule involving Reflex Angle Bearings (4:59)
- Video - Finding Partial Lengths using Sine Rule (4:31)
- Video - Sine Rule Applications (7:34)
- Video - Sine Rule involving Similar Triangles (3:02)

#### Arc Length in Degrees

- Video - Length of an Arc (7:59)
- Video - Arc of a Circle (2:29)
- Video - Radius and Arc of a Circle (3:24)
- Video - Perimeter of Circles and Composite Shapes (4:50)
- Video - Finding Length of an Arc (1:33)
- Video - Finding the Radius using Arc Length (1:53)
- Video - Perimeters of Sectors using Arc Length (2:52)
- Video - Perimeters of Shapes involving Rectangles and Circles (2:58)
- Video - Perimeters of Composite Shapes involving Circles (4:42)
- Video - Arc Length of a Sector Formula (2:02)
- Video - Finding Arc Length of a Sector (5:04)
- Video - Applications of Length of an Arc (5:38)
- Topic - Finding Arc and Perimeter of a Sector 1
- Topic - Finding Arc and Perimeter of a Sector 2
- Topic - Finding Arc and Perimeter of a Sector 3
- Topic - Finding Arc and Perimeter of a Sector 4
- Topic - Finding Radius of a Sector 1
- Topic - Finding Radius of a Sector 2

#### Sector Area in Radians

- Video - Sector Area with Angles in Radians (1:24)
- Video - Sector Area with Angles in Degrees (1:04)
- Video - Finding Angles using Sector Area (1:03)
- Video - Finding Radius using Sector Area (1:18)
- Video - Sector Area involving Arc Length (2:18)
- Video - Area involving Two Sectors (2:21)
- Video - Area of Semicircles and Sectors (2:26)
- Video - Area of Sectors and Isosceles Triangles (8:34)

#### Time Zones

- Video - Finding Time Differences (6:18)
- Video - Time Zone (6:10)
- Topic - Time Difference using GMT Ahead
- Topic - Time Difference using GMT Behind
- Topic - Time Difference using GMT Ahead and Behind
- Topic - Local Times by Time Zone Same Day Ahead
- Topic - Local Times by Time Zone Same Day Behind
- Topic - Local Times by Time Zone Different Day Ahead
- Topic - Local Times by Time Zone Different Day Behind