geo formulas

Lines & Angles

• Supplementary Angles: a + b = 180^\circ

• Complementary Angles: a + b = 90^\circ

• Vertical Angles: Equal (congruent)

• Parallel Lines Cut by a Transversal:
  

  • Alternate interior angles = congruent

  • Corresponding angles = congruent

  • Same-side interior = supplementary

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Triangles

• Area:
  A = \frac{1}{2} \cdot \text{base} \cdot \text{height}

• Pythagorean Theorem (Right triangle):
  a^2 + b^2 = c^2

• Triangle Sum Theorem:
  a + b + c = 180^\circ

• 45°-45°-90° Triangle:
  x : x : x\sqrt{2}

• 30°-60°-90° Triangle:
  x : x\sqrt{3} : 2x

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Circles

• Circumference:
  C = 2\pi r \text{ or } \pi d

• Area:
  A = \pi r^2

• Arc Length:
  \frac{\theta}{360} \cdot 2\pi r

• Sector Area:
  \frac{\theta}{360} \cdot \pi r^2

• Equation of a Circle:
  (x - h)^2 + (y - k)^2 = r^2

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Quadrilaterals

• Area of a Rectangle/Square:
  A = \text{base} \cdot \text{height}

• Area of a Trapezoid:
  A = \frac{1}{2} (b_1 + b_2) \cdot h

• Area of a Parallelogram:
  A = b \cdot h

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3D Figures (Volume & Surface Area)

Prism & Cylinder:

• Volume:
  V = B \cdot h

• Surface Area of Cylinder:
  SA = 2\pi r^2 + 2\pi rh

Pyramid & Cone:

• Volume:
  V = \frac{1}{3} B h

• Surface Area of Cone:
  SA = \pi r^2 + \pi r l

Sphere:

• Volume:
  V = \frac{4}{3} \pi r^3

• Surface Area:
  SA = 4\pi r^2

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Coordinate Geometry

• Distance Formula:
  \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

• Midpoint Formula:
  \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

• Slope Formula:
  \frac{y_2 - y_1}{x_2 - x_1}