Reference Sheet
Core Concepts & Formulas to Memorize
📐 Angles & Lines
Complementary: two angles summing to 90°
Supplementary: two angles summing to 180°
Vertical angles: opposite angles at an intersection — always equal
Parallel lines cut by transversal: alternate interior angles are equal; co-interior angles sum to 180°
Supplementary: two angles summing to 180°
Vertical angles: opposite angles at an intersection — always equal
Parallel lines cut by transversal: alternate interior angles are equal; co-interior angles sum to 180°
Comp
a + b = 90°
Supp
a + b = 180°
🔺 Triangles
Angle sum: all three angles sum to 180°
Pythagorean Theorem: for right triangles, a² + b² = c² where c is the hypotenuse
Special right triangles: 30-60-90 → sides 1 : √3 : 2; 45-45-90 → sides 1 : 1 : √2
Pythagorean Theorem: for right triangles, a² + b² = c² where c is the hypotenuse
Special right triangles: 30-60-90 → sides 1 : √3 : 2; 45-45-90 → sides 1 : 1 : √2
a² + b² = c² | Area = ½ × base × height
30-60-90
1 : √3 : 2
45-45-90
1 : 1 : √2
Example
A right triangle has legs of 6 and 8. Find the hypotenuse.
6² + 8² = 36 + 64 = 100 → c = 10
⭕ Circles
Circumference: C = 2πr Area: A = πr²
Arc length: (θ/360°) × 2πr Sector area: (θ/360°) × πr²
Inscribed angle: equals half the central angle that intercepts the same arc
Arc length: (θ/360°) × 2πr Sector area: (θ/360°) × πr²
Inscribed angle: equals half the central angle that intercepts the same arc
C = 2πr | A = πr² | Arc = (θ/360)·2πr
Example
A circle has radius 5. Find its area and circumference.
Area = π(5)² = 25π | C = 2π(5) = 10π
⬜ Quadrilaterals & Polygons
Rectangle: Area = l × w, Perimeter = 2(l + w)
Parallelogram: Area = base × height
Trapezoid: Area = ½(b₁ + b₂) × h
Interior angle sum of n-gon: (n − 2) × 180°
Parallelogram: Area = base × height
Trapezoid: Area = ½(b₁ + b₂) × h
Interior angle sum of n-gon: (n − 2) × 180°
Sum of interior angles = (n − 2) × 180°
📏 Coordinate Geometry
Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂−y₁)/(x₂−x₁)
Parallel lines: equal slopes Perpendicular: slopes multiply to −1
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂−y₁)/(x₂−x₁)
Parallel lines: equal slopes Perpendicular: slopes multiply to −1
d = √[(Δx)² + (Δy)²] | m₁ × m₂ = −1 (perpendicular)
📦 3D Solids
Rectangular prism: V = l×w×h, SA = 2(lw + lh + wh)
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Sphere: V = (4/3)πr³, SA = 4πr²
Cone: V = (1/3)πr²h
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Sphere: V = (4/3)πr³, SA = 4πr²
Cone: V = (1/3)πr²h
Cylinder V = πr²h | Sphere V = (4/3)πr³
🔁 Similarity & Congruence
Similar triangles (AA, SAS, SSS): corresponding sides are proportional, corresponding angles are equal
Congruent triangles (SSS, SAS, ASA, AAS, HL): all corresponding sides and angles are equal
Scale factor k: areas scale by k², volumes by k³
Congruent triangles (SSS, SAS, ASA, AAS, HL): all corresponding sides and angles are equal
Scale factor k: areas scale by k², volumes by k³
If scale factor = k → Area ratio = k² → Volume ratio = k³
📐 Trigonometry Basics
SOH-CAH-TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Pythagorean identity: sin²θ + cos²θ = 1
Common values: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 90° = 1
Pythagorean identity: sin²θ + cos²θ = 1
Common values: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 90° = 1
sin 30°
1/2
cos 30°
√3/2
sin 45°
√2/2
tan 45°
1
Practice Exam
20 Essential Geometry Problems
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