Geometrical terms

1. Overview

Geometrical terms are the foundation for understanding shapes, space, and spatial relationships. This topic covers the essential vocabulary for describing and working with geometrical figures, lines, angles, and solids. Mastering these terms is crucial for success in geometry problems, construction, navigation, and engineering applications within the IGCSE Cambridge Mathematics (0580) syllabus. This revision note will equip you with the definitions, concepts, and problem-solving skills needed to confidently tackle geometrical term questions.

---

Key Definitions

• Point: A precise location in space, usually represented by a dot and a capital letter.
• Vertex: A corner where two or more lines meet (plural: vertices).
• Line: A straight path that extends infinitely in both directions. In IGCSE, "line" usually refers to a line segment with two endpoints.
• Plane: A flat, two-dimensional surface that extends infinitely.
• Parallel: Two lines that are always the same distance apart and never meet (denoted by arrows: >>).
• Perpendicular: Two lines that meet at a 90° angle.
• Perpendicular Bisector: A line that cuts another line segment exactly in half at a 90° angle.
• Bearing: A horizontal angle measured clockwise from North, expressed as three digits (e.g., 045°).
• Similar: Shapes that are the same shape but different sizes (angles are equal, sides are proportional).
• Congruent: Shapes that are identical in both shape and size.
• Scale Factor: The ratio by which a shape is enlarged or reduced.

---

Core Content

Vocabulary of a Circle

You must be able to identify and name the following parts of a circle:

• Radius: The distance from the center to the edge.
• Diameter: The distance across the circle through the center ($d = 2r$).
• Circumference: The distance around the outside of the circle.
• Arc: A portion of the circumference.
• Chord: A straight line joining two points on the circumference.
• Tangent: A straight line that touches the circle at exactly one point.
• Sector: A "pizza slice" area bounded by two radii and an arc.
• Segment: An area bounded by a chord and an arc.

Worked example 1 — Circle Diameter

Question: A circle has a radius of 7 cm. Calculate the diameter.

1. Identify the relationship: $d = 2 \times r$ (Diameter is twice the radius)
2. Substitute values: $d = 2 \times 7 \text{ cm}$
3. Calculate: $d = 14 \text{ cm}$
4. Final Answer: $\boxed{14 \text{ cm}}$

Worked example 2 — Circle Radius

Question: The diameter of a circle is 24 cm. What is the radius?

1. Identify the relationship: $d = 2 \times r$ (Diameter is twice the radius)
2. Rearrange for r: $r = \frac{d}{2}$ (Divide both sides by 2)
3. Substitute values: $r = \frac{24 \text{ cm}}{2}$
4. Calculate: $r = 12 \text{ cm}$
5. Final Answer: $\boxed{12 \text{ cm}}$

---

Extended Content (Extended Only)

Types of Angles

• Acute: $0^\circ < \theta < 90^\circ$
• Right angle: Exactly $90^\circ$
• Obtuse: $90^\circ < \theta < 180^\circ$
• Reflex: $180^\circ < \theta < 360^\circ$

Polygons and Triangles

• Triangles: Equilateral (all sides/angles equal), Isosceles (two sides/angles equal), Scalene (no sides equal).
• Quadrilaterals: Square, Rectangle, Parallelogram, Rhombus, Trapezium, Kite.
• Regular Polygon: All sides and interior angles are equal.

Interior and Exterior Angles

• Interior angles: The angles inside a polygon.
• Exterior angles: The angles between a side of a polygon and an extension of the adjacent side. They always sum to $360^\circ$.

Worked example 3 — Interior Angles

Question: Find the sum of the interior angles of a hexagon ($n = 6$).

1. Formula: $\text{Sum} = (n - 2) \times 180^\circ$ (Sum of interior angles formula)
2. Substitute: $\text{Sum} = (6 - 2) \times 180^\circ$
3. Simplify: $\text{Sum} = 4 \times 180^\circ$
4. Calculation: $\text{Sum} = 720^\circ$
5. Final Answer: $\boxed{720^\circ}$

Worked example 4 — Exterior Angle of a Regular Polygon

Question: Calculate the size of one exterior angle of a regular pentagon.

1. Formula: $\text{Exterior angle} = \frac{360^\circ}{n}$ (Exterior angle of a regular polygon)
2. Substitute: $\text{Exterior angle} = \frac{360^\circ}{5}$ (A pentagon has 5 sides)
3. Calculate: $\text{Exterior angle} = 72^\circ$
4. Final Answer: $\boxed{72^\circ}$

Bearings

Bearings must follow three rules:

1. Measured from North.
2. Measured Clockwise.
3. Written as three figures (e.g., $005^\circ$).

Worked example 5 — Reverse Bearings

Question: The bearing of B from A is $070^\circ$. Find the bearing of A from B.

1. **Draw a sketch**: Place North arrows at both A and B. 2. **Identify the angle**: The bearing of B from A is $070^\circ$. 3. **Use parallel line rules**: The North lines are parallel. The angle between the North line at B and the line BA is $70^\circ$ (alternate angles). 4. **Calculate the bearing of A from B**: Since the bearing of B from A is less than $180^\circ$, add $180^\circ$. 5. **Calculation**: $70^\circ + 180^\circ = 250^\circ$ 6. **Final Answer**: $\boxed{250^\circ}$

Worked example 6 — Bearing Calculation with Angles

Question: Town B is 50 km due East of town A. Town C is 70 km from town B on a bearing of $050^\circ$. Find the bearing of B from C.

1. Draw a diagram: Draw points A and B, with B east of A. Draw a North line at B and mark the angle of $50^\circ$ clockwise to point C.
2. Identify angles: The angle between the North line at B and BC is $50^\circ$. The angle between AB and the North line at B is $90^\circ$.
3. Calculate the interior angle at B: The interior angle ABC is $90^\circ + 50^\circ = 140^\circ$.
4. Draw a North line at C: Draw a North line at C. The angle between the North line at B and the North line at C is $0^\circ$ (parallel lines).
5. Calculate the angle between CB and the North line at C: The angle between CB and the North line at C is $50^\circ$ (alternate angles).
6. Calculate the bearing of B from C: Since the angle is less than $180^\circ$, we need to find the reflex angle. $180^\circ + 50^\circ = 230^\circ$.
7. Final Answer: $\boxed{230^\circ}$

Solids and Nets

• Solid: A 3D shape (e.g., Prism, Pyramid, Cylinder, Cone, Sphere).
• Net: A 2D pattern that can be folded to make a 3D solid.

---

Key Equations

• Diameter of a circle: $\bf{d = 2r}$
• Sum of interior angles of a polygon: $\bf{(n - 2) \times 180^\circ}$ (where $n$ is the number of sides)
• One interior angle of a regular polygon: $\bf{\frac{(n - 2) \times 180^\circ}{n}}$
• Sum of exterior angles: $\bf{360^\circ}$
• Exterior angle of a regular polygon: $\bf{\frac{360^\circ}{n}}$
• Scale Factor ($k$): $\bf{k = \frac{\text{New Length}}{\text{Original Length}}}$

Note: The formula for the sum of interior angles of a polygon, $(n-2) \times 180^\circ$, is not usually provided on the formula sheet; you must memorise it.

---

Common Mistakes to Avoid

• ❌ Wrong: Writing a bearing as $60^\circ$.
  • ✓ Right: Always use three digits for bearings: $060^\circ$.
• ❌ Wrong: Measuring a bearing anti-clockwise.
  • ✓ Right: Always measure clockwise from the North line.
• ❌ Wrong: Confusing "The bearing of A from B" with "The bearing of B from A".
  • ✓ Right: Put your protractor (or start your measurement) at the point following the word "from". Visualize yourself standing at that point and measuring the angle to the other point.
• ❌ Wrong: Assuming all 4-sided shapes are rectangles.
  • ✓ Right: Check for parallel symbols and right-angle markers to identify specific quadrilaterals like parallelograms or trapeziums.
• ❌ Wrong: Forgetting to add or subtract $180^\circ$ when finding reverse bearings.
  • ✓ Right: If the original bearing is less than $180^\circ$, add $180^\circ$. If it's greater than $180^\circ$, subtract $180^\circ$. Always sketch a diagram to check.

---

Exam Tips

• Command Words: If a question says "Sketch," it doesn't need to be to scale, but labels must be accurate. If it says "Construct," you must use a compass and ruler accurately.
• Real-world contexts: Bearings often appear in ship or plane navigation questions. Always draw the "North" line at every point mentioned in the journey.
• Protractor Use: In the exam, make sure the $0^\circ$ mark is perfectly aligned with your North line.
• Formula Sheet: Note that the interior angle sum formula $(n-2) \times 180$ is not usually provided; you must memorise it.
• Calculator Tip: When calculating angles using trigonometry (used later in bearings), ensure your calculator is in DEG (Degree) mode, not RAD or GRAD.