5 Geometry and trigonometry
- Syllabus
- 2024
- Topic
- 5
- Level
- —
Learning objectives
Measure an angle from the correct reference line
An angle measures the turn from one line or direction to another. The unit is the degree, written ∘, and the vertex is the point where the two arms meet. A numerical angle is meaningful only when both arms are identified.
full turn=360∘,half turn=180∘,right angle=90∘
To measure an angle, place the protractor centre on the vertex, align its zero line with the stated reference arm, and read the scale where the second arm crosses. Choose the scale that starts at zero on the aligned arm and report the value in degrees.
In ray physics, the angle of incidence is measured between the incoming ray and the normal—the line perpendicular to the surface at the point of incidence. If the ray makes 11∘ with the surface, it makes 90∘−11∘=79∘ with the normal.
Do not measure from whichever nearby line looks convenient. Measuring from the surface instead of the normal gives the complementary angle, and reading the wrong protractor scale can produce a second incorrect value.
Read a three-dimensional object from a two-dimensional view
A three-dimensional object has length, width and height, but a two-dimensional representation can display only two spatial directions at once. Read the labels, view direction and scale to decide which real dimensions and relationships the drawing preserves.
| Representation | What it can show directly |
|---|---|
| plan view | length and width as seen from above |
| front or side elevation | one horizontal dimension and height |
| perspective sketch | the object's overall 3D form, but apparent lengths may be distorted |
| scale diagram | measurable lengths linked to real lengths by a stated factor |
If a scale diagram states 1cm on the page represents 10cm in the laboratory, a measured height of 3.1cm represents 3.1×10=31cm. Apply the scale factor to a length measured in the same direction, then attach the real unit.
Match corresponding corners, edges and faces across views; a feature hidden in one view may be visible in another. A cross-section describes a slice through the object rather than its outside silhouette.
A labelled physics sketch is not automatically drawn to scale. Never infer a real length, angle or proportion from appearance alone unless dimensions or a scale are supplied; perspective can make equal or parallel edges look unequal or convergent.
Choose area, surface area or volume—and keep the units
Area measures a two-dimensional region, surface area totals the areas of an object's outside faces, and volume measures the three-dimensional space it occupies. The requested quantity determines both the formula and the power of the unit.
| Shape or object | Calculation | Unit type |
|---|---|---|
| rectangle | A=lw | square units |
| triangle | A=21bh | square units |
| cube, side a | S=6a2 | square units |
| cube, side a | V=a3 | cubic units |
Use a triangle's perpendicular height, not necessarily a sloping side. Convert every length to the same unit before multiplying. For a cube, one face is a square of area a2 and there are six equal faces; volume multiplies three perpendicular dimensions.
A cube with side 4cm has surface area 6(42)=96cm2 and volume 43=64cm3. The numbers differ because surface area counts six faces, while volume fills the interior. An irregular contact area on 2cm grid squares can be estimated by counting complete and combined partial squares, then multiplying by 4cm2 per square.
Do not report area in cm or volume in cm2. Squaring or cubing the numerical length also squares or cubes its unit; converting after the calculation without applying that power changes the physical quantity.