Geometry
Geometry
What you'll learn
- Find angle sums in polygons and quadrilaterals
- State and use properties of parallelograms, rectangles, rhombuses, squares, and trapeziums
- Construct quadrilaterals given sufficient measurements
- Identify faces, edges, and vertices of 3-D shapes and use Euler's formula
Key concepts
Understanding Quadrilaterals
Angle Sum Properties
Polygon angle sum formula: Sum of interior angles of an n-sided polygon = (n − 2) × 180°
| Polygon | Sides (n) | Angle sum |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
| Octagon | 8 | 1080° |
Sum of exterior angles of any convex polygon = 360° (one exterior angle per vertex)
Worked Example: Three angles of a quadrilateral are 75°, 110°, and 95°. Find the fourth. Sum = 360° → Fourth angle = 360° − (75 + 110 + 95) = 360° − 280° = 80°
Properties of Quadrilaterals
Hierarchy: Square ⊂ Rectangle ⊂ Parallelogram and Square ⊂ Rhombus ⊂ Parallelogram
Parallelogram:
| Property | Detail |
|---|---|
| Opposite sides | Equal and parallel |
| Opposite angles | Equal |
| Consecutive angles | Supplementary (add to 180°) |
| Diagonals | Bisect each other |
If one angle of a parallelogram = 70°, the adjacent angle = 110°, opposite angle = 70°.
Rectangle:
| Property | Detail |
|---|---|
| All parallelogram properties | Apply |
| All angles | 90° each |
| Diagonals | Equal in length AND bisect each other |
Rhombus:
| Property | Detail |
|---|---|
| All parallelogram properties | Apply |
| All sides | Equal |
| Diagonals | Perpendicular bisectors of each other |
| Diagonals | Bisect the vertex angles |
Area of rhombus = (d₁ × d₂) / 2, where d₁ and d₂ are the diagonals.
Square:
| Property | Detail |
|---|---|
| All rectangle AND rhombus properties | Apply |
| All sides | Equal |
| All angles | 90° |
| Diagonals | Equal, perpendicular, bisect each other at 90°, bisect vertex angles at 45° |
Trapezium:
| Property | Detail |
|---|---|
| One pair of opposite sides | Parallel (called bases) |
| Sum of co-interior angles | 180° (between the parallel sides) |
| Diagonals | Generally unequal and do NOT bisect each other |
Isosceles trapezium: Non-parallel sides are equal; base angles are equal; diagonals are equal.
Comparison table:
| Shape | All sides equal | All angles 90° | Diagonals equal | Diagonals ⊥ | Diagonals bisect |
|---|---|---|---|---|---|
| Parallelogram | No | No | No | No | Yes |
| Rectangle | No | Yes | Yes | No | Yes |
| Rhombus | Yes | No | No | Yes | Yes |
| Square | Yes | Yes | Yes | Yes | Yes |
| Trapezium | No | No | No | No | No |
Construction of Quadrilaterals
A quadrilateral has 5 independent measurements (from its 4 sides and 2 diagonals or angles). You need at least 5 to construct it uniquely.
Cases:
| Given | Elements |
|---|---|
| Case 1 | Four sides + one diagonal |
| Case 2 | Four sides + one angle |
| Case 3 | Three sides + two diagonals |
| Case 4 | Three sides + two included angles |
| Case 5 | Special quadrilateral (e.g. rhombus: 1 side + 1 diagonal) |
Worked Example — Case 1: Construct quadrilateral PQRS with PQ = 5 cm, QR = 4 cm, RS = 6 cm, SP = 3.5 cm, and diagonal PR = 6.5 cm. Steps:
- Draw PQ = 5 cm.
- With P as centre and radius 6.5 cm, and Q as centre and radius 4 cm, intersect to find R.
- With P as centre and radius 3.5 cm, and R as centre and radius 6 cm, intersect to find S.
- Join all vertices.
Constructing a parallelogram: Opposite sides are equal, so only 3 measurements needed (2 adjacent sides + 1 angle, or 2 sides + diagonal).
Worked Example — Rhombus: Side = 5 cm, diagonal = 8 cm. Draw diagonal AC = 8 cm. With A and C as centres and radius 5 cm each, draw arcs intersecting at B (above) and D (below). Join ABCD.
Visualising Solid Shapes
Faces, Edges, Vertices
| Solid | Faces (F) | Edges (E) | Vertices (V) |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square pyramid | 5 | 8 | 5 |
| Triangular pyramid (tetrahedron) | 4 | 6 | 4 |
| Cone | 2 | 1 (curved) | 1 |
| Cylinder | 3 | 2 (circular) | 0 |
| Sphere | 1 | 0 | 0 |
Euler's Formula
For any convex polyhedron:
F + V − E = 2
| Shape | F | V | E | F + V − E |
|---|---|---|---|---|
| Cube | 6 | 8 | 12 | 6+8−12 = 2 ✓ |
| Triangular prism | 5 | 6 | 9 | 5+6−9 = 2 ✓ |
| Square pyramid | 5 | 5 | 8 | 5+5−8 = 2 ✓ |
Worked Example: A solid has 7 faces and 10 vertices. How many edges does it have? F + V − E = 2 → 7 + 10 − E = 2 → E = 15
Nets of Solids
A net is a 2-D shape that folds to make a 3-D solid.
| Solid | Net description |
|---|---|
| Cube | 6 equal squares in a cross or T-shape |
| Cuboid | 6 rectangles (3 pairs of equal rectangles) |
| Square pyramid | 1 square + 4 triangles around it |
| Triangular prism | 2 triangles + 3 rectangles |
Views of 3-D Shapes
- Front view (elevation): seen from the front
- Side view: seen from the left or right
- Top view (plan): seen from above
A cube seen from any direction looks like a square. A cylinder from the front looks like a rectangle; from the top, a circle.
Quick check
- Find the value of x: angles of a quadrilateral are 3x, 2x, 4x, and (x+20)°.
- The diagonals of a rhombus are 10 cm and 24 cm. Find the side of the rhombus.
- A solid has 12 edges and 6 faces. Use Euler's formula to find the vertices.
- State TWO properties that a square has but a rectangle does not.
- Draw the net of a triangular prism and label all faces.
Open the Practice tab for graded questions on Geometry.
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