Distance formula: d = √((x₂-x₁)² + (y₂-y₁)²). It comes from the Pythagorean theorem.

SAS: if two sides and the included angle of one triangle equal those of another, the triangles are congruent.

A midsegment connects midpoints of two sides. It is parallel to the third side and half its length.

A full rotation = 360°. Half rotation = 180° (straight angle). Quarter rotation = 90° (right angle).

Area = a × b. Use Pythagorean theorem to find b: b = √(d²-a²). Then A = a√(d²-a²).

Circumcenter = intersection of perpendicular bisectors. It is equidistant from all three vertices.

A tangent line touches the circle at exactly one point. It is always perpendicular to the radius at the point of tangency.

An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

Volume of cone = (1/3)πr²h. It's one-third of the cylinder with the same base and height.

Volume of triangular prism = base area × height (length). Base area = ½bh of the triangular cross-section.

Regular octagon interior angle = (8-2)×180°/8 = 1080°/8 = 135°.

Two tangent segments drawn from an external point to a circle are always equal in length.

Sum = (8-2) × 180° = 6 × 180° = 1080°.

Sum = (6-2) × 180° = 4 × 180° = 720°.

Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter.

Arc measure equals the central angle in degrees. A 60° central angle intercepts a 60° arc.

In a right triangle, if altitude h is drawn to hypotenuse: h² = (segment₁)(segment₂) — the geometric mean.

Arc length s = rθ, where θ is in radians and r is the radius. This is the most elegant arc length formula.

A point equidistant from both sides of an angle lies on the angle bisector.

Scale factor = ratio of corresponding side lengths. Area ratio = (scale factor)². Volume ratio = (scale factor)³.