Geometry Fundamentals
Master geometry from basic angle relationships to 3D volume calculations with this comprehensive flashcard deck. Geometry is one of the oldest branches of mathematics and is heavily tested on the SAT, ACT, GCSE, A-Level, and state standardized exams. A strong geometry foundation is also essential for physics, architecture, engineering, and computer graphics. This deck covers all essential geometry formulas and theorems: angle types and relationships, triangle properties and congruence rules, the Pythagorean theorem, polygon properties, circle theorems, area and perimeter formulas, and 3D shape volumes and surface areas. Based on publicly available educational resources including OpenStax Geometry and CK-12.
All Cards in This Deck
Question
What are complementary and supplementary angles?
Answer
Complementary: Two angles that add to 90°. Supplementary: Two angles that add to 180°. Memory trick: C comes before S in the alphabet, 90 comes before 180.
Question
What are vertical angles?
Answer
When two lines intersect, the angles directly across from each other are vertical angles — they are always equal. The adjacent angles are supplementary.
Question
What are alternate interior angles?
Answer
When a transversal crosses two parallel lines, alternate interior angles (on opposite sides of the transversal, between the parallel lines) are equal.
Question
What are co-interior (same-side interior) angles?
Answer
When a transversal crosses two parallel lines, co-interior angles (on the same side, between the parallel lines) are supplementary — they add up to 180°.
Question
What is the Pythagorean theorem?
Answer
In a right triangle: a² + b² = c² where c is the hypotenuse (longest side, opposite the right angle). Example: 3² + 4² = 5² (the 3-4-5 right triangle).
Question
What are common Pythagorean triples?
Answer
Sets of whole numbers satisfying a²+b²=c²: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 7-24-25. Multiples of these also work (e.g., 9-12-15).
Question
What are the triangle congruence rules?
Answer
SSS (all three sides equal), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), RHS (right angle + hypotenuse + side).
Question
What are the triangle similarity rules?
Answer
AA (two angles equal — most common), SSS (all sides in proportion), SAS (two sides in proportion + included angle equal). Similar triangles have equal angles and proportional sides.
Question
What is the sum of interior angles of a polygon?
Answer
(n−2) × 180° where n = number of sides. Triangle = 180°. Quadrilateral = 360°. Pentagon = 540°. Hexagon = 720°.
Question
What is the exterior angle of a regular polygon?
Answer
360° ÷ n where n = number of sides. The sum of all exterior angles of any convex polygon = 360°. Example: Regular hexagon exterior angle = 60°.
Question
What is the area of a triangle?
Answer
A = ½ × base × height. Also: A = ½ab sin C (using two sides and included angle). Also: Heron's formula A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
Question
What is the area of a circle?
Answer
A = πr² where r is the radius. Circumference = C = 2πr = πd where d is the diameter. π ≈ 3.14159.
Question
What is the area of a trapezoid (trapezium)?
Answer
A = ½ × (a + b) × h where a and b are the parallel sides (bases) and h is the perpendicular height between them.
Question
What is the volume of a cylinder?
Answer
V = πr²h where r = radius of base, h = height. Surface area = 2πr² + 2πrh (two circles + curved surface).
Question
What is the volume of a cone?
Answer
V = ⅓πr²h. Slant height l = √(r²+h²). Surface area = πr² + πrl (base + curved surface).
Question
What is the volume of a sphere?
Answer
V = (4/3)πr³. Surface area = 4πr².
Question
What is the volume of a pyramid?
Answer
V = ⅓ × base area × height. For a square pyramid: V = ⅓ × s² × h.
Question
What are the circle theorems?
Answer
1. Angle at centre = 2 × angle at circumference (same arc). 2. Angles in same segment are equal. 3. Angle in semicircle = 90°. 4. Opposite angles in cyclic quadrilateral add to 180°. 5. Tangent is perpendicular to radius at point of contact.
Question
What is the arc length formula?
Answer
Arc length = (θ/360°) × 2πr for angle θ in degrees. In radians: Arc length = rθ. A full circle arc (θ=360°) = full circumference = 2πr.
Question
What is the sector area formula?
Answer
Sector area = (θ/360°) × πr² for angle θ in degrees. In radians: Sector area = ½r²θ.
Question
What is the equation of a circle?
Answer
(x−h)² + (y−k)² = r² where (h,k) is the centre and r is the radius. Standard form centred at origin: x² + y² = r².
Question
What is the midpoint theorem for triangles?
Answer
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
Question
What is the centroid of a triangle?
Answer
The point where all three medians intersect. It is located ⅔ of the way from each vertex to the opposite midpoint. Also the triangle's centre of mass.
Question
What are special right triangles?
Answer
30-60-90: Sides in ratio 1 : √3 : 2. The side opposite 30° is shortest, opposite 60° is √3 times the shortest. 45-45-90: Sides in ratio 1 : 1 : √2.
Question
What is the surface area of a cube vs rectangular prism?
Answer
Cube: SA = 6s² (6 identical square faces). Rectangular prism (cuboid): SA = 2(lw + lh + wh) where l=length, w=width, h=height.