Keystone Reference Geometry Formula Sheet

Basic Geometric Definitions

• Point: A location in space with no size or dimension.
• Line: A collection of points extending infinitely in two directions.
• Line Segment: A part of a line bounded by two endpoints.
• Ray: A part of a line that starts at an endpoint and extends infinitely in one direction.
• Plane: A flat surface that extends infinitely in all directions.
• Angle: The figure formed by two rays with a common endpoint (vertex).

Angles and Their Properties

• Acute Angle: An angle measuring between 0 and 90.
• Right Angle: An angle measuring exactly 90.
• Obtuse Angle: An angle measuring between 90 and 180.
• Straight Angle: An angle measuring exactly 180.

• Complementary Angles: Two angles that sum to 90.
• Supplementary Angles: Two angles that sum to 180.
• Vertical Angles: Opposite angles formed by intersecting lines; they are congruent.
• Corresponding Angles: Angles in the same relative position at each intersection where a straight line crosses two others.
• Alternate Interior/Exterior Angles: Angles formed when a line crosses two other lines.

Triangle Properties and Formulas

• Equilateral Triangle: All sides equal, all angles equal (60 each).
• Isosceles Triangle: At least two sides equal, base angles equal.
• Scalene Triangle: No sides equal, no angles equal.
• Right Triangle: One angle equals 90.

• For right triangles, where a and b are the legs and c is the hypotenuse.

• Where b is the base and h is the height.

• 45-45-90 Triangle: Legs are equal, hypotenuse = leg 2
• 30-60-90 Triangle: Shorter leg = x, Longer leg = x3, Hypotenuse = 2x

• AA Similarity: If two angles of one triangle are congruent to two angles of another triangle.
• SSS Similarity: If corresponding sides are proportional.
• SAS Similarity: If two sides are proportional and the included angle is congruent.

• SSS Congruence: If three sides of one triangle are congruent to three sides of another.
• SAS Congruence: If two sides and the included angle of one triangle are congruent to another.
• ASA Congruence: If two angles and the included side of one triangle are congruent to another.
• AAS Congruence: If two angles and a non-included side of one triangle are congruent to another.
• HL Congruence: If the hypotenuse and leg of one right triangle are congruent to another.

Circle Properties and Formulas

• Radius (r): Distance from center to any point on the circle.
• Diameter (d): Distance across the circle through the center (d = 2r).
• Chord: A line segment whose endpoints lie on the circle.
• Sector: Region bounded by two radii and an arc.
• Secant: A line that intersects a circle at two points.
• Tangent: A line that touches a circle at exactly one point.

• Circumference: C = 2r = d
• Area: A = r
• Arc Length: s = (/360) 2r, where is the central angle
• Sector Area: A = (/360) r, where is the central angle

Polygon Properties and Formulas

• Sum of Interior Angles: (n-2) 180, where n is the number of sides
• Measure of One Interior Angle: ((n-2) 180) n
• Measure of One Exterior Angle: 360 n

• Parallelogram: Opposite sides parallel and equal, opposite angles equal.
• Rectangle: Parallelogram with four right angles.
• Rhombus: Parallelogram with four equal sides.
• Square: Rectangle with four equal sides.
• Trapezoid: Quadrilateral with exactly one pair of parallel sides.

• Rectangle: A = l w (length width)
• Square: A = s (side)
• Parallelogram: A = b h (base height)
• Trapezoid: A = (1/2) h (b + b)

• Rectangle: P = 2l + 2w
• Square: P = 4s
• Trapezoid: P = sum of all sides

3D Geometric Solids

• Prism: A solid with two parallel, congruent polygonal bases and rectangular faces.
• Pyramid: A solid with a polygonal base and triangular faces that meet at a common vertex.
• Cylinder: A solid with two parallel circular bases and a curved surface.
• Cone: A solid with a circular base and a curved surface that tapers to a point.
• Sphere: A perfectly round solid where all points on the surface are equidistant from the center.

• Rectangular Prism: V = l w h
• Triangular Prism: V = (1/2) b h l
• Cylinder: V = rh
• Pyramid: V = (1/3) base area h
• Cone: V = (1/3) rh
• Sphere: V = (4/3)r

• Rectangular Prism: SA = 2lw + 2lh + 2wh
• Cylinder: SA = 2r + 2rh
• Cone: SA = r + rl (where l is the slant height)
• Sphere: SA = 4r

Coordinate Geometry

• Lines are parallel if they have the same slope (m = m).

• Lines are perpendicular if their slopes are negative reciprocals (m m = -1).

Important Geometry Theorems

• The Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• The Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
• The Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
• The Midsegment Theorem: The segment joining the midpoints of two sides of a triangle is parallel to the third side and is half as long.
• Tangent-Secant Theorem: If a tangent and a secant intersect at a point outside a circle, then the square of the length of the tangent segment equals the product of the lengths of the secant segment and its external part.