Essential Geometry Formulas
Geometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. Mastering geometry formulas is fundamental for solving a wide range of mathematical problems and real-world applications in architecture, engineering, design, and many other fields.
Basic Plane Geometry
Rectangle
Area = length width
A = l w
Perimeter = 2 (length + width)
P = 2(l + w)
Square
Area = side
A = s
Perimeter = 4 side
P = 4s
Diagonal = side 2
d = s2
Triangle
Area = base height
A = bh
Perimeter = sum of all sides
P = a + b + c
c = a + b
Parallelogram
Area = base height
A = bh
Perimeter = 2 (side + side)
P = 2(a + b)
Trapezoid
Area = (base + base) height
A = (b + b)h
Perimeter = base + base + side + side
P = b + b + s + s
Circle
Area = radius
A = r
Circumference = 2 radius
C = 2r
Arc Length = (angle/360) 2r
For arc of degrees
Sector Area = (angle/360) r
For sector of degrees
Ellipse
Area = semi-major axis semi-minor axis
A = ab
Perimeter [3(a+b) - ((3a+b)(a+3b))]
Solid Geometry
Rectangular Prism
Volume = length width height
V = l w h
Surface Area = 2(lengthwidth + lengthheight + widthheight)
SA = 2(lw + lh + wh)
Cube
Volume = side
V = s
Surface Area = 6 side
SA = 6s
Space Diagonal = side 3
d = s3
Sphere
Volume = (4/3) radius
V = (4/3)r
Surface Area = 4 radius
SA = 4r
Cylinder
Volume = radius height
V = rh
Surface Area = 2r + 2rh
SA = 2r(r + h)
Lateral Surface Area = 2rh
Area of the curved surface only
Cone
Volume = (1/3) radius height
V = (1/3)rh
Surface Area = r + r (r + h)
SA = r(r + (r + h))
Slant Height = (r + h)
l = (r + h)
Pyramid
Volume = (1/3) base area height
V = (1/3)Bh
Surface Area = Base Area + (1/2) base perimeter slant height
SA = B + (1/2)Pl
Triangle Theorems
Triangle Properties
Sum of angles in a triangle = 180
Triangle Inequality Theorem:
The sum of any two sides of a triangle must be greater than the third side.
Similar Triangles
If two triangles are similar, corresponding angles are congruent and corresponding sides are proportional.
AA Similarity: Two angles of one triangle are congruent to two angles of another triangle.
SSS Similarity: Corresponding sides of the triangles are in proportion.
SAS Similarity: Two sides are in proportion, and the included angles are congruent.
Congruent Triangles
SSS Congruence: Three sides of one triangle are congruent to three sides of another.
SAS Congruence: Two sides and the included angle of one triangle are congruent to those of another.
ASA Congruence: Two angles and the included side of one triangle are congruent to those of another.
AAS Congruence: Two angles and a non-included side of one triangle are congruent to those of another.
HL Congruence (Right Triangles): The hypotenuse and a leg of one right triangle are congruent to those of another.
Triangle Centers
Centroid: The point where the three medians intersect.
Circumcenter: The point where the perpendicular bisectors of the sides intersect.
Incenter: The point where the angle bisectors intersect.
Orthocenter: The point where the altitudes intersect.
Coordinate Geometry
Distance Formula
Distance between two points: d = [(x-x) + (y-y)]
Midpoint Formula
Midpoint of a segment: ((x+x)/2, (y+y)/2)
Slope Formula
Slope of a line: m = (y-y)/(x-x)
Equation of a Line
Slope-Intercept Form: y = mx + b
Point-Slope Form: y - y = m(x - x)
Standard Form: Ax + By = C
Parallel and Perpendicular Lines
Parallel lines have equal slopes.
Perpendicular lines have slopes that are negative reciprocals (product = -1).
Circle Theorems
Circle Properties
Central Angle Theorem: The measure of a central angle equals the measure of its intercepted arc.
Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
Thales' Theorem: An angle inscribed in a semicircle is a right angle.
Chord Theorems
In the same circle or congruent circles, congruent chords have congruent arcs and congrent central angles.
The perpendicular from the center of a circle to a chord bisects the chord.
Chord-Chord Product Theorem: If two chords intersect, the products of the segments of each chord are equal.
Tangent Lines
A tangent to a circle is perpendicular to the radius at the point of tangency.
Tangent-Segment Theorem: From a point outside a circle, the two tangent segments to the circle are congruent.
Secant-Tangent Theorem: The square of a tangent segment equals the product of the secant segment and its external part.
Polygon Properties
Regular Polygons
Sum of Interior Angles = (n-2) 180
Where n is the number of sides
Measure of One Interior Angle = (n-2) 180/n
Sum of Exterior Angles = 360
For any convex polygon
Measure of One Exterior Angle = 360/n
Area = (1/2) apothem perimeter
A = (1/2)ap
Number of Diagonals = n(n-3)/2
Special Theorems and Formulas
Heron's Formula
Area of a triangle = [s(s-a)(s-b)(s-c)]
Where s = (a+b+c)/2 is the semi-perimeter
Law of Sines
a/sin A = b/sin B = c/sin C
For any triangle with sides a, b, c opposite angles A, B, C
Law of Cosines
c = a + b - 2ab cos C
For any triangle with sides a, b, c opposite angles A, B, C
Pick's Theorem
Area = I + B/2 - 1
For a simple polygon with vertices on lattice points, where I = interior points and B = boundary points
Euler's Formula for Polyhedra
V - E + F = 2
Where V = vertices, E = edges, F = faces
