VCE Further Mathematics Units 3 and 4 Geometry and Measurement
Course

VCE Further Mathematics Units 3 and 4 – Geometry and Measurement

5.1 Surface Area
5.2 Similarity
5.3 Trigonometry
5.4 Non-Right Angled Triangles
5.5 Spherical Geometry

30 Lessons

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

SURFACE AREA

Surface Area of Prisms

Surface Area of Pyramids

Surface Area of Cylinders

Surface Area of Spheres

Surface Area of Cones

Surface Area of Composite Solids

SIMILARITY

Scale Factor

Similar Triangles

Proof of Similar Triangles

Scale Drawings

Area of Similar Figures

Volume of Similar Figures

TRIGONOMETRY

Angles of Elevation and Angle of Depression

True Bearings

Obtuse Angles

Three Dimensional Trigonometry

NON-RIGHT ANGLED TRIANGLES

Area of Triangles using Sides and Angles

Sine Rule

Applications of Sine Rule

SPHERICAL GEOMETRY

Arc Length in Degrees

Arc Length in Radians

Sector Area in Radians

Area of Minor Segments

Latitude and Longitude

Time Zones