◆ Mathematics — Geometry

Geometry Master Quiz

20 Essential Problems Across All Core Topics

20 Questions
30 Minutes
Exam-Style
📐 Concept Review

Core Concepts & Formulas

Tap each unit to expand concepts, key formulas to memorize, and worked examples.

Unit 1

Angles & Lines

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Supplementary
∠A + ∠B = 180°
Angles on a straight line
Complementary
∠A + ∠B = 90°
Angles in a right angle
Vertical Angles
∠A = ∠C, ∠B = ∠D
Opposite angles equal
Parallel Lines
Alt. int. = Corr. = equal
Co-int. angles sum to 180°
⭐ Must Memorize
  • Angles in a triangle sum to 180°
  • Angles around a point sum to 360°
  • Alternate interior angles are equal (parallel lines)
  • Co-interior (same-side) angles are supplementary
  • Exterior angle = sum of two non-adjacent interior angles
Worked Example

Two parallel lines are cut by a transversal. One alternate interior angle is 65°. Find the co-interior angle.

Co-interior angle = 180° − 65° = 115°
Unit 2

Triangles

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Area
A = ½ × b × h
base × height ÷ 2
Pythagorean
a² + b² = c²
Right triangle only
Perimeter
P = a + b + c
Sum of all sides
Angle Sum
∠A + ∠B + ∠C = 180°
All triangles
⭐ Must Memorize
  • Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
  • Isoceles triangle: base angles equal
  • Equilateral: all angles = 60°
  • Triangle inequality: any side < sum of other two sides
  • Area of equilateral: (√3/4)s²
Worked Example

A right triangle has legs 6 and 8. Find the hypotenuse.

c = √(6² + 8²) = √(36 + 64) = √100 = 10
Unit 3

Quadrilaterals & Polygons

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Rectangle
A = l × w, P = 2(l+w)
Parallelogram
A = b × h
h = perpendicular height
Trapezoid
A = ½(a+b) × h
a,b = parallel sides
Polygon Angles
Sum = (n−2)×180°
n = number of sides
⭐ Must Memorize
  • Each interior angle of regular polygon: (n−2)×180° ÷ n
  • Each exterior angle of regular polygon: 360° ÷ n
  • Sum of exterior angles (any convex polygon): 360°
  • Rhombus area: d₁ × d₂ ÷ 2 (diagonals)
Worked Example

Find each interior angle of a regular hexagon.

Sum = (6−2)×180° = 720°. Each angle = 720° ÷ 6 = 120°
Unit 4

Circles

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Circumference
C = 2πr = πd
Area
A = πr²
Arc Length
L = (θ/360°) × 2πr
Sector Area
A = (θ/360°) × πr²
⭐ Must Memorize
  • Central angle = intercepted arc
  • Inscribed angle = ½ × intercepted arc
  • Angle in a semicircle = 90°
  • Tangent ⊥ radius at point of tangency
  • Two tangents from external point: equal length
Worked Example

A circle has radius 5 cm. A sector has central angle 72°. Find its area. (Use π ≈ 3.14)

A = (72/360) × π × 5² = (1/5) × 78.5 = 15.7 cm²
Unit 5

Surface Area & Volume

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Cube
V = s³, SA = 6s²
Cuboid
V = lwh, SA = 2(lw+lh+wh)
Cylinder
V = πr²h, SA = 2πr(r+h)
Cone
V = ⅓πr²h, SA = πr(r+l)
Sphere
V = (4/3)πr³, SA = 4πr²
Pyramid
V = ⅓ × base area × h
⭐ Must Memorize
  • Cone slant height: l = √(r² + h²)
  • Prism volume: cross-section area × length
  • If linear scale = k, area scale = k², volume scale = k³
Worked Example

A cylinder has radius 3 and height 7. Find its volume. (Use π ≈ 3.14)

V = π × 3² × 7 = π × 63 ≈ 197.82
Unit 6

Coordinate Geometry

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Distance
d = √((x₂−x₁)²+(y₂−y₁)²)
Midpoint
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope
m = (y₂−y₁)/(x₂−x₁)
Line Equation
y = mx + c
⭐ Must Memorize
  • Parallel lines have equal slopes
  • Perpendicular lines: slopes multiply to −1
  • Horizontal line: slope = 0; Vertical line: slope = undefined
Worked Example

Find the midpoint of A(2, 6) and B(8, 4).

M = ((2+8)/2, (6+4)/2) = (5, 5)

Practice Problems

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