Geometry Essentials: Theorems, Formulas, and Shapes

Posted on Sep 23, 2025 in Mathematics

Geometric Foundations: Theorems and Postulates

Triangle Congruence and Similarity

  • Angle-Angle (AA) Similarity Postulate

    If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

  • Side-Angle-Side (SAS) Similarity Theorem

    If an angle of one triangle is congruent to an angle of another triangle, and the sides including these angles are proportional (e.g., AB/DE = AC/DF), then the triangles are similar.

Properties of Geometric Shapes

  • Parallelogram Properties

    • Opposite sides are parallel and congruent.
    • Opposite angles are congruent.
    • Diagonals bisect each other.

  • Proving a Quadrilateral is a Parallelogram

    To prove a quadrilateral is a parallelogram, show one of the following:

    • Both pairs of opposite sides are parallel.
    • Both pairs of opposite sides are congruent.
    • Both pairs of opposite angles are congruent.
    • The diagonals bisect each other.
    • One pair of opposite sides is both congruent and parallel.

    Note on parallel lines: This often involves identifying alternate interior angles (Z-angles), same-side interior angles (C-angles), or corresponding angles (F-angles).

  • Rectangle Properties

    • Has four right angles.
    • Diagonals are congruent.

  • Rhombus Properties

    • Consecutive sides are congruent (all four sides are congruent).
    • Diagonals are perpendicular.
    • Each diagonal bisects two angles at the corners.

  • Right Triangle Property

    The midpoint of the hypotenuse is equidistant from all three vertices.

Chapter 8: Right Triangles and Trigonometry

Geometric Mean

  • To find the geometric mean between two numbers (e.g., 5 and 11):

    1. Set up the proportion: 5/x = x/11
    2. Solve for x: x² = 55, so x = √55.

  • When an altitude is drawn to the hypotenuse of a right triangle, it creates three similar right triangles. The altitude is the geometric mean between the two segments of the hypotenuse.

Pythagorean Theorem

In a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c².

Special Right Triangles

  • 45-45-90 Triangle

    The hypotenuse is √2 times the length of an equal leg.

    Hypotenuse = √2 × Leg

  • 30-60-90 Triangle

    The hypotenuse is 2 times the length of the shorter leg.

    The longer leg is √3 times the length of the shorter leg.

    Hypotenuse = 2 × Short Leg

    Long Leg = √3 × Short Leg

Basic Trigonometry (SOH CAH TOA)

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Example: If given an angle (e.g., 56°) and the adjacent side (e.g., 5), and you need to find the opposite side (x):

tan(56°) = x / 5

Chapter 11: Area Formulas for Polygons and Circles

Area Formulas for Polygons

  • Rectangle

    Area = base × height (A = b × h)

  • Parallelogram

    Area = base × height (A = b × h)

    Note: An altitude may need to be drawn to find the height, potentially using trigonometry or special right triangles.

  • Triangle

    Area = (base × height) / 2 (A = ½ b × h)

  • Rhombus

    Area = (diagonal 1 × diagonal 2) / 2 (A = ½ d&sub1; × d&sub2;)

  • Trapezoid

    Area = ½ × height × (base 1 + base 2) (A = ½ h (b&sub1; + b&sub2;))

Apothem and Regular Polygons

The apothem is the distance from the center to the midpoint of a side of a regular polygon. It forms right triangles with the radius and half-side.

  • Central Angle Calculation

    Central Angle = 360° / number of sides

  • Apothem Triangles for Regular Polygons

    • Equilateral Triangle (3 sides): Central angle = 120°. Forms 30-60-90 right triangles.
    • Square (4 sides): Central angle = 90°. Forms 45-45-90 right triangles.
    • Regular Hexagon (6 sides): Central angle = 60°. Forms 30-60-90 right triangles.
    • Regular Pentagon (5 sides): Central angle = 72°. Forms right triangles with angles 54°, 36°, 90°. (May require trigonometry to solve).

Circles: Circumference, Area, Arc Length, and Sector Area

  • Circumference

    Circumference = 2πr

  • Area

    Area = πr²

  • Arc Length

    Formula: (Central Angle / 360°) × 2πr

    • Example 1: Radius = 5, Arc AB = 60°.
    • Fraction of circle = 60/360 = 1/6.
    • Length of Arc AB = (1/6) × (2π × 5) = 5π/3.
    • Example 2: Radius = 9, Angle = 120°.
    • Fraction of circle = 120/360 = 1/3.
    • Length of Arc = (1/3) × (2π × 9) = 6π.
    • Major Arc: To find the major arc length (greater than 180°), subtract the given minor angle from 360°.
    • Example: If minor angle = 120°, major angle = 360° – 120° = 240°.
    • Length of Major Arc = (240/360) × (2π × 9) = (2/3) × (18π) = 12π.

  • Sector Area

    Formula: (Central Angle / 360°) × πr²

    • Example 1: Radius = 5, Sector AOB = 60°.
    • Fraction of circle = 60/360 = 1/6.
    • Area of Sector AOB = (1/6) × (π × 5²) = 25π/6.
    • Example 2: Radius = 9, Sector AOB = 120°.
    • Fraction of circle = 120/360 = 1/3.
    • Area of Sector AOB = (1/3) × (π × 9²) = 27π.

Chapter 9: Circle Theorems and Properties

Tangent Theorems

  • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency, forming a right angle.
  • Two tangent segments drawn to a circle from the same external point are congruent.
  • If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.

Arc and Chord Theorems

  • Congruent arcs have congruent chords.
  • Congruent chords have congruent arcs.

Angle Relationships in Circles

  • The measure of an inscribed angle is half the measure of its intercepted arc.
  • The measure of a central angle is equal to the measure of its intercepted arc.
  • The measure of an angle formed by a tangent and a chord drawn to the point of tangency is half the measure of its intercepted arc.

Segment Relationships in Circles

  • Tangent-Secant Segment Theorem

    If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external segment.

Chapter 12: Volume and Surface Area of Prisms

Right Prism Formulas

  • Lateral Area (LA)

    The lateral area of a right prism equals the perimeter of its base (P) multiplied by the height of the prism (h).

    Formula: LA = P × h

    To find P: Add up all the side lengths of the base.

  • Volume (V)

    The volume of a right prism equals the area of its base (B) multiplied by the height of the prism (h).

    Formula: V = B × h

  • Example: Right Trapezoidal Prism Calculations

    Consider a right trapezoidal prism where you need to find the Lateral Area, Area of Base, and Volume.

    • A. Lateral Area (LA):
      • Add up all the side lengths of the trapezoidal base (e.g., P = 28 units).
      • Multiply the perimeter by the height of the prism (e.g., h = 10 units).
      • LA = 28 × 10 = 280 square units.
    • B. Area of Base (B):
      • Since the base is a trapezoid, use the trapezoid area formula: B = ½ × hbase × (b&sub1; + b&sub2;).
    • C. Volume (V):
      • Once the area of the base (B) is found, multiply it by the prism’s height (h).
      • V = B × h.