📐 Geometry — Formulas
Essential formulas and shortcuts for working with geometry in MATHCOUNTS.
Triangle Formulas
Basic Properties
Sum of angles: ∠A + ∠B + ∠C = 180° Triangle inequality: a + b > c, b + c > a, a + c > b Area: A = (1/2) × base × height Perimeter: P = a + b + c
Pythagorean Theorem
Right triangles: a² + b² = c² where c is hypotenuse Distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Example: If legs are 3 and 4, hypotenuse is √(3² + 4²) = 5
Special Right Triangles
30°-60°-90°: Sides in ratio 1 : √3 : 2
- If short leg = x, then long leg = x√3, hypotenuse = 2x
- If hypotenuse = x, then short leg = x/2, long leg = x√3/2
45°-45°-90°: Sides in ratio 1 : 1 : √2
- If leg = x, then hypotenuse = x√2
- If hypotenuse = x, then leg = x/√2 = x√2/2
3-4-5 triangle: Sides in ratio 3 : 4 : 5 5-12-13 triangle: Sides in ratio 5 : 12 : 13 8-15-17 triangle: Sides in ratio 8 : 15 : 17
Heron’s Formula
Area: A = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2 Semi-perimeter: s = (a + b + c)/2
Example: Triangle with sides 3, 4, 5
- s = (3 + 4 + 5)/2 = 6
- A = √[6(6-3)(6-4)(6-5)] = √[6×3×2×1] = √36 = 6
Quadrilateral Formulas
Rectangle
Area: A = length × width Perimeter: P = 2(length + width) Diagonal: d = √(length² + width²)
Example: Rectangle 6×4 has area 24, perimeter 20, diagonal √52
Square
Area: A = side² Perimeter: P = 4 × side Diagonal: d = side√2
Example: Square with side 5 has area 25, perimeter 20, diagonal 5√2
Parallelogram
Area: A = base × height Perimeter: P = 2(a + b) Properties: Opposite sides equal and parallel, opposite angles equal
Example: Parallelogram with base 8 and height 5 has area 40
Rhombus
Area: A = (1/2) × diagonal₁ × diagonal₂ Perimeter: P = 4 × side Properties: All sides equal, diagonals perpendicular and bisect each other
Example: Rhombus with diagonals 6 and 8 has area 24
Trapezoid
Area: A = (1/2) × (base₁ + base₂) × height Perimeter: P = sum of all sides Properties: One pair of parallel sides
Example: Trapezoid with bases 6 and 10, height 4 has area 32
Circle Formulas
Basic Circle
Circumference: C = 2πr = πd Area: A = πr² Diameter: d = 2r Radius: r = d/2
Example: Circle with radius 5 has circumference 10π and area 25π
Arc and Sector
Arc length: s = rθ (where θ is in radians) Sector area: A = (1/2)r²θ (where θ is in radians) Degree conversion: θ (radians) = θ (degrees) × π/180
Example: 60° sector of radius 6 has arc length 2π and area 6π
Chord Properties
Chord length: c = 2√(r² - d²) where d is distance from center Distance from center: d = √(r² - (c/2)²) Power of a point: AB × BC = DB × BE
Example: Chord 8 units long, 3 units from center: radius = √(4² + 3²) = 5
Tangent Properties
Tangent length: t = √(d² - r²) where d is distance from point to center Power of a point: t² = s × e where s is secant length, e is external part Two tangents: From external point, two tangents are equal length
Example: Tangent 6 units long, secant 9 units long: external part = 9 - 6 = 3
Similarity and Congruence
Similarity
Scale factor: k = ratio of corresponding sides Area ratio: A₁/A₂ = k² Volume ratio: V₁/V₂ = k³ Perimeter ratio: P₁/P₂ = k
Example: If scale factor is 2, area ratio is 4, volume ratio is 8
Congruence Tests
SSS: All three sides equal SAS: Two sides and included angle equal ASA: Two angles and included side equal AAS: Two angles and non-included side equal HL: Hypotenuse and leg equal (right triangles only)
Area Formulas
Basic Shapes
Rectangle: A = length × width Square: A = side² Triangle: A = (1/2) × base × height Parallelogram: A = base × height Trapezoid: A = (1/2) × (base₁ + base₂) × height Circle: A = πr²
Composite Figures
Method: Break into basic shapes, find area of each, add together Subtraction method: Find area of larger shape, subtract area of smaller shape
Example: L-shaped figure = rectangle + triangle
Perimeter Formulas
Basic Shapes
Rectangle: P = 2(length + width) Square: P = 4 × side Triangle: P = a + b + c Circle: C = 2πr
Composite Figures
Method: Add all outer sides Subtraction method: Find perimeter of larger shape, subtract inner perimeters
Example: Rectangle with hole = 2(l + w) + 2πr (where r is hole radius)
Volume Formulas
Basic Shapes
Rectangular prism: V = length × width × height Cube: V = side³ Cylinder: V = πr²h Cone: V = (1/3)πr²h Sphere: V = (4/3)πr³ Pyramid: V = (1/3) × base area × height
Composite Solids
Method: Break into basic shapes, find volume of each, add together Subtraction method: Find volume of larger solid, subtract volume of smaller solid
Example: L-shaped solid = rectangular prism + triangular prism
Surface Area Formulas
Basic Shapes
Rectangular prism: SA = 2(lw + lh + wh) Cube: SA = 6 × side² Cylinder: SA = 2πr² + 2πrh Cone: SA = πr² + πrl Sphere: SA = 4πr²
Composite Solids
Method: Find surface area of each part, add together Subtraction method: Find surface area of larger solid, subtract inner surface areas
Example: Cylinder with hole = 2πr² + 2πrh - 2πr² (where r is hole radius)
Coordinate Geometry
Distance Formula
Two points: d = √[(x₂ - x₁)² + (y₂ - y₁)²] Point to line: d = |ax + by + c|/√(a² + b²)
Example: Distance between (1, 2) and (4, 6) = √[(4-1)² + (6-2)²] = 5
Midpoint Formula
Two points: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) Line segment: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Example: Midpoint of (1, 2) and (4, 6) = (2.5, 4)
Slope Formula
Two points: m = (y₂ - y₁)/(x₂ - x₁) Parallel lines: m₁ = m₂ Perpendicular lines: m₁ × m₂ = -1
Example: Slope of line through (1, 2) and (4, 6) = 4/3
Equation of a Line
Slope-intercept: y = mx + b Point-slope: y - y₁ = m(x - x₁) Standard form: Ax + By = C Two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Example: Line through (1, 2) with slope 3: y - 2 = 3(x - 1)
Transformations
Translations
Notation: (x, y) → (x + a, y + b) Example: Translate (2, 3) by (1, 4) → (3, 7)
Reflections
Over x-axis: (x, y) → (x, -y) Over y-axis: (x, y) → (-x, y) Over y = x: (x, y) → (y, x) Over y = -x: (x, y) → (-y, -x)
Example: Reflect (2, 3) over x-axis → (2, -3)
Rotations
90° counterclockwise: (x, y) → (-y, x) 180°: (x, y) → (-x, -y) 270° counterclockwise: (x, y) → (y, -x) 90° clockwise: (x, y) → (y, -x)
Example: Rotate (2, 3) 90° counterclockwise → (-3, 2)
Dilations
Notation: (x, y) → (kx, ky) Center at origin: (x, y) → (kx, ky) Center at (a, b): (x, y) → (a + k(x - a), b + k(y - b))
Example: Dilate (2, 3) by factor 2 → (4, 6)
Mental Math Shortcuts
Common Areas
Rectangle 6×4: Area = 24, Perimeter = 20 Square side 5: Area = 25, Perimeter = 20, Diagonal = 5√2 Circle radius 5: Circumference = 10π, Area = 25π Triangle 3-4-5: Area = 6, Perimeter = 12
Common Volumes
Cube side 4: Volume = 64, Surface area = 96 Cylinder r=3, h=5: Volume = 45π, Surface area = 48π Sphere radius 3: Volume = 36π, Surface area = 36π
Common Ratios
30°-60°-90°: 1 : √3 : 2 45°-45°-90°: 1 : 1 : √2 3-4-5: 3 : 4 : 5 5-12-13: 5 : 12 : 13
Common Applications
Real-world Problems
Architecture: Use geometry in building design Engineering: Use geometry in structural design Art: Use geometry in artistic composition Sports: Use geometry in field design and equipment
Problem-solving Strategies
Draw diagrams: Visualize the problem Identify given information: List what you know Use appropriate formulas: Apply the right formula Check your answer: Verify the result makes sense
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