📐 Complete K-12 Geometry Formulas 📐
Basic 2D Shapes
Square
Area = s²
Perimeter = 4s
Diagonal = s√2
s = side length
Rectangle
Area = l × w
Perimeter = 2(l + w)
Diagonal = √(l² + w²)
l = length, w = width
Parallelogram
Area = b × h
Perimeter = 2(a + b)
b = base, h = height, a = side
Rhombus
Area = (d₁ × d₂) / 2
Area = b × h
Perimeter = 4s
d₁, d₂ = diagonals, b = base, h = height, s = side
Trapezoid
Area = ((a + b) / 2) × h
Perimeter = a + b + c + d
a, b = parallel sides, h = height, c, d = other sides
Kite
Area = (d₁ × d₂) / 2
Perimeter = 2(a + b)
d₁, d₂ = diagonals, a, b = sides
Triangles
General Triangle
Area = (b × h) / 2
Perimeter = a + b + c
Heron's Formula: A = √(s(s-a)(s-b)(s-c))
b = base, h = height, a,b,c = sides, s = (a+b+c)/2
Equilateral Triangle
Area = (s²√3) / 4
Perimeter = 3s
Height = (s√3) / 2
s = side length
Isosceles Triangle
Area = (b × h) / 2
Perimeter = 2a + b
Height = √(a² - (b²/4))
a = equal sides, b = base, h = height
Right Triangle
Area = (a × b) / 2
Pythagorean: c² = a² + b²
Perimeter = a + b + c
a, b = legs, c = hypotenuse
Circles
Circle
Area = πr²
Circumference = 2πr = πd
Diameter = 2r
r = radius, d = diameter, π ≈ 3.14159
Sector (Pie Slice)
Area = (θ/360) × πr²
Arc Length = (θ/360) × 2πr
θ = angle in degrees, r = radius
Segment
Area = (r²/2)(θ - sin θ)
θ = angle in radians, r = radius
Ellipse
Area = πab
Perimeter ≈ π(a + b)
a = semi-major axis, b = semi-minor axis
3D Solids (Volume & Surface Area)
Cube
Volume = s³
Surface Area = 6s²
Space Diagonal = s√3
s = side length
Rectangular Prism (Box)
Volume = l × w × h
Surface Area = 2(lw + lh + wh)
Diagonal = √(l² + w² + h²)
l = length, w = width, h = height
Sphere
Volume = (4/3)πr³
Surface Area = 4πr²
r = radius
Cylinder
Volume = πr²h
Surface Area = 2πr² + 2πrh
Lateral Area = 2πrh
r = radius, h = height
Cone
Volume = (1/3)πr²h
Surface Area = πr² + πrl
Slant Height: l = √(r² + h²)
r = radius, h = height, l = slant height
Pyramid
Volume = (1/3) × B × h
Surface Area = B + (1/2)Pl
B = base area, h = height, P = base perimeter, l = slant height
Prism
Volume = B × h
Surface Area = 2B + Ph
B = base area, h = height, P = base perimeter
Torus (Donut)
Volume = 2π²Rr²
Surface Area = 4π²Rr
R = major radius, r = minor radius
Hemisphere
Volume = (2/3)πr³
Surface Area = 3πr²
r = radius
Coordinate Geometry
Distance Formula
d = √((x₂-x₁)² + (y₂-y₁)²)
(x₁,y₁) and (x₂,y₂) are two points
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
M = midpoint of two points
Slope of a Line
m = (y₂-y₁)/(x₂-x₁)
Rise over Run
m = slope
Equation of a Line
y = mx + b (Slope-intercept)
y - y₁ = m(x - x₁) (Point-slope)
Ax + By = C (Standard)
m = slope, b = y-intercept
Parallel & Perpendicular Lines
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = -1
m = slope
Trigonometry
Basic Trig Ratios
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
θ = angle in right triangle
Reciprocal Functions
csc θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
θ = angle
Pythagorean Identity
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Law of Sines
a/sin A = b/sin B = c/sin C
a,b,c = sides; A,B,C = opposite angles
Law of Cosines
c² = a² + b² - 2ab cos C
a,b,c = sides; C = angle opposite side c
Triangle Area (Trig)
Area = (1/2)ab sin C
a,b = two sides; C = included angle
Special Formulas & Theorems
Triangle Angle Sum
A + B + C = 180°
Sum of angles in any triangle
Polygon Interior Angles
Sum = (n - 2) × 180°
Each angle = ((n-2) × 180°)/n
n = number of sides (regular polygon)
Polygon Exterior Angles
Sum = 360°
Each angle = 360°/n
n = number of sides (regular polygon)
Arc Measures
Arc measure = Central angle
Inscribed angle = (1/2) Arc
In a circle
Regular Polygon
Area = (1/2) × P × a
P = n × s
P = perimeter, a = apothem, n = sides, s = side length
Similar Figures
Area ratio = k²
Volume ratio = k³
k = scale factor (linear ratio)