Basic Geometric Formulas and Properties

Points, Lines, and Angles

Fundamental Concepts

• Point: An exact location with no size.
• Line: An infinite set of points extending in two directions. Defined by a linear equation y = mx + b in the Cartesian plane.
• Segment: Part of a line bounded by two endpoints.
• Ray: Part of a line that starts at an endpoint and extends infinitely in one direction.
• Angle: Formed by two rays sharing a common endpoint (the vertex). Measured in degrees () or radians (rad).

Angle Relationships

Triangles

Area

For a triangle with baseb and heighth:

Area = (bh) / 2

Heron's Formula

When only the three side lengths a, b, c are known:

Semiperimeter s = (a + b + c) / 2

Area = [s(sa)(sb)(sc)]

Special Triangles

• Equilateral: All sides equal; each interior angle 60; area = (3/4)a.
• Isosceles: Two sides equal; altitude from the vertex bisects the base.
• Right: Pythagorean theorem: a + b = c (c is the hypotenuse).

Quadrilaterals

Rectangle

Area = length width

Perimeter = 2(length + width)

Square

All sides equal; a special rectangle.
Area = side
Perimeter = 4 side

Parallelogram

Opposite sides are parallel and equal.
Area = base height (height measured perpendicular to the base).
Perimeter = 2(base + side)

Rhombus

All sides equal; opposite angles are equal. Two common area formulas:

• Area = (diagonal diagonal) / 2
• Area = base height (same as a parallelogram)

Trapezoid (US) / Trapezium (UK)

Only one pair of parallel sides (bases).
Area = ( (base + base) / 2 ) height

Circles

Key Quantities

• Radius (r): Distance from center to any point on the circumference.
• Diameter (d): Twice the radius, d = 2r.
• Circumference: Length around the circle, C = 2r = d.
• Area: A = r.

Sector and Segment

Sector (portion bounded by two radii and an arc):
Area = (/360) r where is the central angle in degrees.

Segment (area between a chord and the corresponding arc):
Area = (r/2)( sin) with in radians.

Solid Figures (ThreeDimensional)

Cube

• Surface area = 6 side
• Volume = side

Rectangular Prism

• Surface area = 2(lw + lh + wh)
• Volume = length width height

Cylinder

• Base radius = r, height = h
• Surface area = 2r(r + h)
• Volume = rh

Cone

• Base radius = r, height = h, slant height = l = (r + h)
• Surface area = r(r + l)
• Volume = (1/3)rh

Sphere

• Radius = r
• Surface area = 4r
• Volume = (4/3)r

Important Geometric Properties

Congruence and Similarity

Congruent figures have identical size and shape. Corresponding sides and angles are equal.

Similar figures have the same shape but different sizes. Corresponding angles are equal, and side lengths are proportional.

Midpoint Formula (Coordinate Geometry)

For endpoints (x, y) and (x, y):

Midpoint = ((x + x)/2, (y + y)/2)

Distance Formula

Distance between two points (x, y) and (x, y):

Distance = [(xx) + (yy)]

Area of a Polygon (Shoelace Formula)

For a simple polygon with vertices (x,y), (x,y), , (x,y):

Area = | (xy xy) |, where indices wrap around (x = x, y = y).

Eulers Formula for Polyhedra

For any convex polyhedron: V E + F = 2, where V = vertices, E = edges, F = faces.