Geometry Formulas and Other Important Stuff You Should Know

Triangle stuff

• Sum of the interior angles of a triangle: 180º
• Area:
  
• Hero’s area formula:
  
  , where a, b, and c are the lengths of the triangle’s sides and
  
  (S is the semiperimeter, half the perimeter)
• Area of an equilateral triangle:
  
  , where s is a side of the triangle
• The Pythagorean Theorem: a² + b² + c², where a and b are the legs of a right triangle and c is the hypotenuse
• Common Pythagorean triples (side lengths in right triangles):
  • 3–4–5
  • 5–12–13
  • 7–24–25
  • 8–15–17
• Ratios of the sides in special right triangles:
  • The sides opposite the angles in a 45º–45º–90º triangle are in the ratio of
    
    .
  • The sides opposite the angles in a 30º–60º–90º triangle are in the ratio of
    
    .
• Altitude-on-Hypotenuse Theorem: If an altitude is drawn to the hypotenuse of a right triangle, then
  • The two triangles formed are similar to the given triangle and to each other:
    
  • h² = xy
  • h² = xy and b² = xc

Polygon stuff

• Area formulas:
  • Parallelogram: Area = base height
  • Rectangle: Area = base × height
  • Kite or rhombus:
    
  • Square:
    
  • Trapezoid:
    
  • Regular polygon:
    
• Sum of the interior angles in an n-sided polygon: Sum_{Interior angles} = (n – 2)180º
• Measure of each interior angle of a regular (or other equiangular) n-sided polygon:
  
  or
  
  (the supplement of an exterior angle)
• Sum of the exterior angles (one at each vertex) of any polygon: Sum_{Exterior angles} = 360º
• Measure of each exterior angle of a regular (or other equiangular) n-sided polygon:
  
• Number of diagonals that can be drawn in an n-sided polygon:
  

Circle stuff

• Circumference: C = 2πr or πd, where r is the radius of the circle and d is its diameter
• Area: Area_Circle = πr²
• Arc length: The length of an arc (part of the circumference) is equal to the circumference of the circle (2πr) times the fraction of the circle represented by the arc.
  
• Sector area: The area of a sector (a pizza-slice shape cut out of a circle) is equal to the area of the circle (πr²) times the fraction of the circle represented by the sector.
  
• Measure of an angle . . .
  • On a circle:
    
  • Inside a circle:
    
  • Outside a circle:
    
• Chord-Chord Power Theorem: When two chords of a circle intersect, the product of the parts of one chord is equal to the product of the parts of the other chord.
• Tangent-Secant Power Theorem: When a tangent and a secant of a circle meet at an external point, the measure of the tangent squared is equal to the product of the secant’s external part and its total length.
• Secant-Secant Power Theorem: When two secants of a circle meet at an external point, the product of one secant’s external part and its total length is equal to the product of the other secant’s external part and its total length.

3-D geometry stuff

• Flat-top objects (prisms and cylinders):
  • Volume: Vol_Flat-top = area_base × height
  • Surface area: SA_Flat-top = 2 × area_base + area_{lateral rectangles} For a cylinder, the single lateral rectangle that wraps around the cylinder has a length equal to the circumference of the cylinder’s base; its width is the cylinder’s height.
• Pointy-top objects (pyramids and cones):
  • Volume:
    
  • Surface area: SA_Pointy-top = area_base + area_{lateral triangles} For a pyramid, the base of each lateral triangle is a side of the pyramid’s base, and the height of the triangle is the slant height of the pyramid. For a cone, one “triangle” wraps around the cone; its base is equal to the circumference of the cone’s base, and its height is equal to the slant height of the cone.
• Sphere:
  • Volume:
    
  • Surface area: SA_Sphere = 4πr²

Coordinate geometry stuff

• Slope formula: Given two points (x₁, y₁) and (x₂, y₂), the slope of the line that goes through the points is
  
  It doesn’t matter which point is designated as (x₁, y₁) and which is designated as (x₂, y₂).
  • The slopes of parallel lines are equal.
  • The slopes of perpendicular lines are opposite reciprocals of each other.
• Midpoint formula: Given a segment with endpoints (x₁, y₁) and (x₂, y₂), the coordinates of its midpoint are
  
  It doesn’t matter which point is (x₁, y₁) and which is(x₂, y₂).
• Distance formula: Given two points (x₁, y₁) and (x₂, y₂), the distance between the points is
  
  It doesn’t matter which point is (x₁, y₁) and which is (x₂, y₂).
• Equations of a line:
  • Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept
  • Point-slope form: y – y₁ = m(x – x₁), where m is the slope and (x₁, y₁) is a point on the line
  • Horizontal line: y = b, where b is the y-intercept
  • Vertical line: x = a, where a is the x-intercept
• Equation of a circle: (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and r is its radius