Geometry: Shapes & Angles,polygons) GL ASSESSMENT QUESTIONS 11 plus exam part -8
11+ GL ASSESSMENT GEOMETRY: ESSENTIAL STUDY GUIDE
PART 1: 2D SHAPES - MUST KNOW PROPERTIES
TRIANGLES
EQUILATERAL TRIANGLE
Sides: All 3 sides EQUAL
Angles: All 3 angles = 60° each
Symmetry:
Lines of symmetry: 3
Rotational symmetry: Order 3
Perimeter: Side × 3
Area: (√3/4) × side²
Key fact: Height < side length
ISOSCELES TRIANGLE
Sides: 2 equal sides, 1 different
Angles: 2 equal angles (base angles), 1 different (vertex angle)
Symmetry:
Lines of symmetry: 1 (through vertex)
Rotational symmetry: Order 1 (none)
Key formula: Vertex angle + 2 × base angle = 180°
Perimeter: (2 × equal side) + base
SCALENE TRIANGLE
Sides: All 3 sides DIFFERENT
Angles: All 3 angles DIFFERENT
Symmetry: NO lines of symmetry
Can be: Acute, right-angled, or obtuse
RIGHT-ANGLED TRIANGLE
Angles: Contains exactly one 90° angle
Sides: Longest side = hypotenuse (opposite right angle)
Pythagoras: a² + b² = c² (where c is hypotenuse)
Area: ½ × base × height
QUADRILATERALS
SQUARE
Sides: All 4 sides EQUAL
Angles: All 4 angles = 90°
Symmetry:
Lines of symmetry: 4 (2 through midpoints, 2 through vertices)
Rotational symmetry: Order 4
Diagonals: Equal, bisect at 90°, bisect angles
Perimeter: 4 × side
Area: side²
RECTANGLE
Sides: Opposite sides equal, adjacent sides different
Angles: All 4 angles = 90°
Symmetry:
Lines of symmetry: 2 (through midpoints)
Rotational symmetry: Order 2
Diagonals: Equal, bisect each other (not at 90°)
Perimeter: 2 × (length + width)
Area: length × width
PARALLELOGRAM
Sides: Opposite sides equal and parallel
Angles: Opposite angles equal, adjacent supplementary
Symmetry:
Lines of symmetry: Usually 0 (except rhombus/rectangle)
Rotational symmetry: Order 2
Diagonals: Bisect each other (not equal, not at 90°)
Area: base × height (NOT side × side)
RHOMBUS
Sides: All 4 sides EQUAL
Angles: Opposite angles equal (not necessarily 90°)
Symmetry:
Lines of symmetry: 2 (through opposite vertices)
Rotational symmetry: Order 2
Diagonals: Perpendicular (90°), bisect each other, bisect angles
Area: ½ × diagonal1 × diagonal2
TRAPEZIUM (UK) / TRAPEZOID (US)
Sides: Exactly ONE pair of parallel sides
Angles: No specific requirements
Symmetry: Usually 0 (unless isosceles trapezium)
Area: ½ × (sum of parallel sides) × height
Isosceles trapezium: Non-parallel sides equal, has 1 line of symmetry
KITE
Sides: Two pairs of ADJACENT sides equal
Angles: One pair of opposite angles equal
Symmetry:
Lines of symmetry: 1 (through unequal angles)
Rotational symmetry: Order 1 (none)
Diagonals: Perpendicular, one diagonal bisected
Area: ½ × diagonal1 × diagonal2
POLYGONS
REGULAR POLYGONS
All sides equal, all angles equal
Sum of interior angles: (n - 2) × 180° (n = number of sides)
Each interior angle: (n - 2) × 180° ÷ n
Sum of exterior angles: ALWAYS 360°
Each exterior angle: 360° ÷ n
Number of diagonals: n(n - 3) ÷ 2
COMMON POLYGONS:
Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6)
Heptagon (7), Octagon (8), Nonagon (9), Decagon (10)
PART 2: 3D SHAPES - ESSENTIAL FACTS
CUBE
Faces: 6 squares
Edges: 12
Vertices: 8
Symmetry: High symmetry in all directions
Volume: side³
Surface area: 6 × side²
CUBOID (RECTANGULAR PRISM)
Faces: 6 rectangles (or squares)
Edges: 12
Vertices: 8
Volume: length × width × height
Surface area: 2(lw + lh + wh)
TRIANGULAR PRISM
Faces: 5 (2 triangles + 3 rectangles)
Edges: 9
Vertices: 6
Volume: area of triangle × length
SQUARE-BASED PYRAMID
Faces: 5 (1 square + 4 triangles)
Edges: 8
Vertices: 5
Volume: ⅓ × base area × height
TRIANGULAR PYRAMID (TETRAHEDRON)
Faces: 4 triangles
Edges: 6
Vertices: 4
Volume: ⅓ × base area × height
CYLINDER
Faces: 3 (2 circles + 1 curved)
Edges: 2 curved edges
Vertices: 0
Volume: Οr²h
Curved surface area: 2Οrh
CONE
Faces: 2 (1 circle + 1 curved)
Edges: 1 curved edge
Vertices: 1
Volume: ⅓Οr²h
SPHERE
Faces: 1 curved surface
Edges: 0
Vertices: 0
Volume: ⁴⁄₃Οr³
Surface area: 4Οr²
EULER'S FORMULA (FOR POLYHEDRA)
F + V = E + 2
Faces + Vertices = Edges + 2
MEMORY TIP: "Faces and Vertices equal Edges plus 2"
PART 3: ANGLES - COMPLETE GUIDE
TYPES OF ANGLES
Acute: < 90°
Right: = 90° (marked with small square)
Obtuse: > 90° but < 180°
Straight line: = 180°
Reflex: > 180° but < 360°
Full turn: = 360°
RELATIONSHIPS
Complementary angles: Sum = 90°
Supplementary angles: Sum = 180°
Adjacent angles: Share vertex and side, no overlap
Vertically opposite: Equal angles where two lines cross
ANGLE RULES - MEMORIZE THESE!
RULE 1: STRAIGHT LINE
Angles on a straight line = 180°
Example: If one angle is 115°, the other is 65°
RULE 2: AROUND A POINT
Angles around a point = 360°
Example: Three angles are 110°, 95°, 80°, fourth = 75°
RULE 3: VERTICALLY OPPOSITE
Vertically opposite angles are EQUAL
Example: Where lines cross, angles opposite each other are equal
RULE 4: TRIANGLE
Sum of angles in triangle = 180°
Corollary:
Right triangle: two acute angles complementary (sum = 90°)
Isosceles: base angles equal
Equilateral: all angles = 60°
RULE 5: QUADRILATERAL
Sum of angles in quadrilateral = 360°
RULE 6: PARALLEL LINES
When parallel lines are crossed by transversal:
Corresponding angles: Equal (F shape)
Alternate angles: Equal (Z shape)
Allied/Co-interior angles: Sum = 180° (C shape)
PART 4: SYMMETRY - KEY POINTS
LINE SYMMETRY (REFLECTION)
Shape can be folded in half to match exactly
Number of lines: Count all possible fold lines
Regular polygon with n sides: n lines of symmetry
ROTATIONAL SYMMETRY
Shape can be rotated and look the same before completing full turn
Order: Number of times shape fits in 360° rotation
Order 1: No rotational symmetry (only looks same at start)
COMMON SHAPES SYMMETRY TABLE
| Shape | Lines of Symmetry | Rotational Symmetry Order |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Parallelogram | 0 (usually) | 2 |
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 |
| Scalene triangle | 0 | 1 |
| Regular pentagon | 5 | 5 |
| Regular hexagon | 6 | 6 |
| Circle | Infinite | Infinite |
| Kite | 1 | 1 |
PART 5: COORDINATES - ESSENTIAL KNOWLEDGE
COORDINATE SYSTEM
(x, y) format
x: Horizontal position (left-right)
y: Vertical position (up-down)
Origin: (0, 0) where axes cross
QUADRANTS
Quadrant I: (+, +) - top right
Quadrant II: (-, +) - top left
Quadrant III: (-, -) - bottom left
Quadrant IV: (+, -) - bottom right
IMPORTANT FORMULAS
DISTANCE BETWEEN TWO POINTS
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Memory tip: "Difference in x squared plus difference in y squared, then square root"
MIDPOINT FORMULA
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Memory tip: "Average the x's, average the y's"
GRADIENT (SLOPE)
Gradient = (y₂ - y₁) ÷ (x₂ - x₁)
Positive slope: line goes up right
Negative slope: line goes down right
Zero slope: horizontal line
Undefined slope: vertical line
TRANSFORMATIONS
REFLECTION
In x-axis: (x, y) → (x, -y) [y changes sign]
In y-axis: (x, y) → (-x, y) [x changes sign]
In line y = x: (x, y) → (y, x) [swap coordinates]
ROTATION (ABOUT ORIGIN)
90° clockwise: (x, y) → (y, -x)
90° anticlockwise: (x, y) → (-y, x)
180°: (x, y) → (-x, -y)
TRANSLATION
Movement by vector (a, b): (x, y) → (x + a, y + b)
PART 6: MEMORY AIDS & EXAM TIPS
ACRONYMS TO REMEMBER
ANGLE TYPES
A ROSE:
A = Acute (< 90°)
R = Right (90°)
O = Obtuse (90°-180°)
S = Straight (180°)
E = everything bigger = Reflex (> 180°)
QUADRILATERAL FAMILY TREE
All Squares are Rectangles but not all Rectangles are Squares
All Squares are Rhombuses but not all Rhombuses are Squares
POLYGON ANGLE FORMULA
SIA: (Sides - 2) × 180 = Interior Angle sum
EXAM TECHNIQUES
DRAW DIAGRAMS: Always sketch the problem
LABEL EVERYTHING: Mark known angles, sides
WORK SYSTEMATICALLY: Use angle rules step by step
CHECK REASONABLENESS: Does your answer make sense?
USE FORMULAS CORRECTLY: Write them down first
COMMON MISTAKES TO AVOID
Confusing area and perimeter: Area = space inside, Perimeter = distance around
Mixing up 2D and 3D properties: Faces vs sides, vertices vs corners
Forgetting units: Always include cm, m, cm², cm³
Misreading scale: Check if diagram is to scale
Assuming symmetry: Don't assume unless stated "regular" or given
QUICK REFERENCE CHART
| Concept | Key Formula/Property | Example |
|---|---|---|
| Triangle angles | Sum = 180° | 60°+70°+50°=180° |
| Quadrilateral angles | Sum = 360° | 90°+100°+80°+90°=360° |
| Regular polygon interior | (n-2)×180°÷n | Octagon: (8-2)×180÷8=135° |
| Regular polygon exterior | 360°÷n | Hexagon: 360÷6=60° |
| Cube volume | side³ | side=4cm, volume=64cm³ |
| Cuboid volume | l×w×h | 3×4×5=60cm³ |
| Distance between points | √[(Ξx)²+(Ξy)²] | (1,1) to (4,5): √(9+16)=5 |
| Midpoint | (avg x, avg y) | (2,4)&(6,8): (4,6) |
PART 7: LAST-MINUTE REVISION CHECKLIST
24 HOURS BEFORE EXAM:
Memorize angle rules (straight line=180°, point=360°)
Know triangle types and properties
Remember quadrilateral hierarchy
Practice 3D shape nets
Review coordinate transformations
1 HOUR BEFORE EXAM:
Write down key formulas
Sketch common shapes with properties
Practice mental calculations
Relax and visualize success
DURING EXAM:
Read questions TWICE
Underline key information
Draw diagrams for geometry questions
Show working (gets method marks)
Check units and reasonableness
Answer ALL questions (no penalty for wrong answers)
REMEMBER: Geometry is visual - if stuck, DRAW IT OUT! Use logical steps, apply rules systematically, and check your work. You've got this!
