Geometry · formula sheet

Geometry: theorems and formulas

Triangles, coordinate geometry, the circle theorems, polygons, and the volume and surface-area formulas for the common solids.

20 formulas and facts in 5 sections

Triangles

Angle sum: $\angle A + \angle B + \angle C = 180^{\circ}$
Area: $\text{Area} = \frac{1}{2}bh$
Pythagorean theorem (right triangles): $a^{2} + b^{2} = c^{2}$
Congruence: Two triangles are congruent by SSS, SAS, ASA, AAS, or HL (right triangles). SSA and AAA do not prove congruence.
Similar triangles with scale factor k: $\frac{\text{side}'}{\text{side}} = k \qquad \frac{\text{Area}'}{\text{Area}} = k^{2}$

Circumference and area: $C = 2\pi r \qquad A = \pi r^{2}$
Inscribed angle: $\text{inscribed angle} = \frac{1}{2}\,(\text{intercepted arc}) = \frac{1}{2}\,(\text{central angle})$
Angle in a semicircle: An inscribed angle that stands on a diameter is 90°.
Tangents: A tangent is perpendicular to the radius at the point of contact. The two tangents drawn from one outside point are equal in length.
Intersecting chords: $a \cdot b = c \cdot d$
a, b and c, d are the two parts of each chord.
Tangent and secant from one point: $(\text{tangent})^{2} = (\text{whole secant}) \times (\text{external part})$

Sum of interior angles (n sides): $(n-2)\cdot 180^{\circ}$
Each exterior angle of a regular polygon: $\frac{360^{\circ}}{n}$
Areas: $\text{parallelogram} = bh \qquad \text{trapezoid} = \frac{1}{2}(b_1 + b_2)h$

Prism and cylinder: $V = Bh \qquad V_{\text{cyl}} = \pi r^{2}h \qquad SA_{\text{cyl}} = 2\pi r^{2} + 2\pi rh$
Pyramid and cone: $V = \frac{1}{3}Bh \qquad V_{\text{cone}} = \frac{1}{3}\pi r^{2}h \qquad \text{lateral area}_{\text{cone}} = \pi r\ell$
B is the area of the base; ℓ is the slant height.
Sphere: $V = \frac{4}{3}\pi r^{3} \qquad SA = 4\pi r^{2}$

A reference for revision. In an exam, use the formula sheet your exam board provides and the notation your course uses.