Geometry
16 concepts
Angles
How far apart two lines (sides) spread from one point. We measure that spread in degrees (°).
▭ S = a × b △ S = c × h ÷ 2Area of Rectangles & Triangles
The amount of flat space a shape covers, counted as how many unit squares fit inside.
▭ P = 2 × (a + b) ⬛ P = 4 × aPerimeter
The total length of the outline once around — just add up all the sides.
V = a × b × cVolume of a Box
The amount of space a solid takes up, counted as how many 1 cm cubes fit inside.
Line Symmetry
The property that folding along a line (the axis) makes the two halves match exactly.
a² + b² = c²The Pythagorean Theorem
In a right triangle, square each of the two shorter sides and add them, and you land exactly on the square of the hypotenuse.
△ABC ≅ △DEFCongruent Triangles
Two triangles being congruent means you can rotate or flip one onto the other perfectly. Every corresponding side and angle is equal.
a : a' = b : b' = c : c' = kSimilarity
Two figures being similar means all corresponding angles are equal and all corresponding sides are the same multiple (k times) of each other.
(n − 2) × 180°Interior & Exterior Angles of Polygons
Add up the interior angles of an n-gon and you get (n−2) × 180°. And the exterior angles of a convex polygon always add up to 360°.
π r² × (θ / 360)Arcs and Sectors of a Circle
A sector is part of a circle. Its central angle θ takes up some fraction of a full turn (360°), and it grabs that same fraction of the circle's total area (or circumference).
d = √((x₂ − x₁)² + (y₂ − y₁)²)Distance in the Coordinate Plane
Find the horizontal gap and the vertical gap between the two points and they form a right triangle; the segment joining the points is its hypotenuse. So the distance is √(horizontal² + vertical²).
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adjTrigonometric Ratios
Pick one acute angle θ in a right triangle, and the ratios between its sides are fixed by the angle alone. Those ratios are sin, cos, and tan.
sin²θ + cos²θ = 1Trigonometric Functions
On a circle of radius 1, after turning by angle θ, the point's vertical position is sinθ and its horizontal position is cosθ. tanθ is their ratio.
(x − a)² + (y − b)² = r²The Equation of a Circle
The points (x,y) whose distance from the center (a,b) is always r form the circle. Writing that condition as an equation gives (x−a)²+(y−b)²=r².
|v| = √(x² + y²)Vectors
A vector is an arrow carrying both size and direction. Moving x sideways and y up is written (x,y).
a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ; |a×b| = |a||b|sinθDot & Cross Products
The dot product a·b is a single number measuring how aligned two vectors are. The cross product a×b is a new vector standing perpendicular to the plane the two vectors span.
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