几何公式

面积公式

Rectangle: P = 2l + 2w—

Square: P = 4s—

Triangle: P = a + b + c—

Circle: C = 2πr = πd—

Parallelogram: P = 2a + 2b—

Trapezoid: P = a + b₁ + c + b₂—

Regular polygon: P = ns—n sides of length s

Semicircle: P = πr + 2r—

体积公式

Rectangle: A = lw—Length × width

Triangle: A = ½bh—Half base × height

Circle: A = πr²—Pi × radius²

Trapezoid: A = ½(b₁+b₂)h—Average bases × height

Parallelogram: A = bh—Base × height

Rhombus: A = ½d₁d₂—Half product of diagonals

Ellipse: A = πab—a, b = semi-axes

Regular polygon: A = ½Pa—P=perimeter, a=apothem

Sector: A = ½r²θ—θ in radians

Heron's: A = √(s(s-a)(s-b)(s-c))—s = (a+b+c)/2

角度

Cube: V = s³—Side cubed

Rectangular prism: V = lwh—Length × width × height

Cylinder: V = πr²h—Base area × height

Sphere: V = (4/3)πr³—Four-thirds pi r-cubed

Cone: V = (1/3)πr²h—Third of cylinder

Pyramid: V = (1/3)Bh—B = base area

Prism: V = Bh—Base area × height

Ellipsoid: V = (4/3)πabc—a, b, c = semi-axes

三角形

Cube: SA = 6s²—6 faces

Rect. prism: SA = 2(lw+lh+wh)—

Cylinder: SA = 2πr²+2πrh—Two circles + lateral

Sphere: SA = 4πr²—

Cone: SA = πr²+πrl—l = slant height

Pyramid: SA = B+½Pl—B=base, P=perim, l=slant

圆

Acute: 0° < θ < 90°—

Right: θ = 90°—

Obtuse: 90° < θ < 180°—

Straight: θ = 180°—

Reflex: 180° < θ < 360°—

Complementary: a+b = 90°—

Supplementary: a+b = 180°—

Vertical angles are equal—

Alternate interior angles =—Parallel lines + transversal

Co-interior angles sum 180°—Same-side interior

Interior angles of polygon—(n-2)×180°

Each angle (regular)—(n-2)×180°/n

Coordinate Geometry

d = √((x₂-x₁)²+(y₂-y₁)²)—Distance formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)—Midpoint formula

m = (y₂-y₁)/(x₂-x₁)—Slope

(x-h)²+(y-k)² = r²—Circle equation, center (h,k)

Ax+By+C=0 distance to point—|Ax₀+By₀+C|/√(A²+B²)

Area = ½|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|—Triangle by coordinates

Transformations

Translation: (x+a, y+b)—Slide

Reflection over x-axis: (x,-y)—

Reflection over y-axis: (-x,y)—

Reflection over y=x: (y,x)—

Rotation 90° CCW: (-y,x)—

Rotation 180°: (-x,-y)—

Rotation 270° CCW: (y,-x)—

Dilation: (kx, ky)—Scale factor k

Circle Theorems

Inscribed angle = ½ central angle—Same arc

Angle in semicircle = 90°—Thales' theorem

Tangent ⊥ radius at point of contact—

Two tangents from external point are equal—

Arc length = rθ—θ in radians

Chord-chord: a×b = c×d—Intersecting chords

Secant-secant: a(a+b) = c(c+d)—From external point

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