- High schoolπGeometryβ‘οΈ
Vector coordinates: AB = (xB β xA, yB β yA)
A vector's coordinates are 'arrival minus departure'. Easy to remember, works everywhere.
- Middle schoolπGeometryπΊ
Volumes of cone, pyramid, and sphere: the 3 formulas
Cone/pyramid = (base Γ height) / 3. Sphere = 4/3Β·ΟΒ·rΒ³. Three formulas for every middle-school solid.
- Middle schoolπGeometryπΊ
A triangle's angles always sum to 180Β°
Why 180Β°? Cut the angles out with scissors β they form a straight line. Proof in 3 animated steps.
- High schoolπGeometryπΊ
Triangle area with sine: Β½Β·aΒ·bΒ·sin(C)
Two sides and the included angle are enough. One formula for any triangle, no height needed.
- Middle schoolπGeometryπΊ
Triangle area: half of the surrounding rectangle
Base Γ height Γ· 2: the formula comes from a rectangle cut in half. The visual proof that makes the formula obvious.
- Middle schoolπGeometryπ
Thales contrapositive: prove two lines are NOT parallel
If ratios aren't equal, lines aren't parallel. The contrapositive of the theorem.
- Middle schoolπGeometryπ
Thales' theorem: when triangles zoom
Two lines from the same point, cut by parallels. The lengths stay proportional. The key to measuring without measuring.
- Middle schoolπGeometryβ‘οΈ
Translation by a vector: the transformation that 'slides' everything
Every point moves by the same vector. No fixed point. Preserves everything (distances, angles, areas, orientation).
- High schoolπGeometryπ
Relative position of two lines: intersecting, parallel, coincident
Compare slopes and y-intercepts to classify 2 lines in 3 seconds.
- Middle schoolπGeometryπ
Pythagoras' theorem, without formulas: the 3-squares proof
A right triangle, three squares drawn on its sides. The visual trick that makes the formula obvious.
- High schoolπGeometryπ
Point-to-line distance: |axβ + byβ + c| / β(aΒ² + bΒ²)
Direct formula for the distance from a point to a line with Cartesian equation ax+by+c = 0.
- Middle schoolπGeometryπΌοΈ
Perimeter or area? Never confuse the two
Perimeter is a length (m, cm). Area is a surface (mΒ², cmΒ²). A simple rule to never mix them up.