NMT / ZNO

Formulas and reference

A compact mathematics reference in two levels: the NMT exam sheet and the full ZNO formula set.

ZNO — full reference · Geometry · 14 formulas
Full ZNO Expanded school formula set. The “NMT” badge marks formulas also present on the compact sheet.
Triangle

Triangle

Sum of angles in a triangle

NMT
\[\alpha+\beta+\gamma=180^\circ\]
7–9 grade

Semiperimeter of a triangle

NMT
\[p=\frac{a+b+c}{2}\]
7–11 grade

Triangle area from a side and altitude

NMT
\[S=\frac12ah_a\]
7–11 grade

Triangle area from two sides and the included angle

NMT
\[S=\frac12bc\sin\alpha\]
9–11 grade

Heron's formula

NMT
\[S=\sqrt{p(p-a)(p-b)(p-c)}\]
9–11 grade

Law of cosines

NMT
\[a^2=b^2+c^2-2bc\cos\alpha\]
9–11 grade

Law of sines

NMT
\[\frac a{\sin\alpha}=\frac b{\sin\beta}=\frac c{\sin\gamma}=2R\]
9–11 grade

Area using the circumradius

\[S=\frac{abc}{4R}\]
9–11 grade

Area using the inradius

\[S=pr\]
8–11 grade

Circumradius

\[R=\frac{abc}{4S}\]
9–11 grade

Inradius

\[r=\frac Sp\]
8–11 grade

Length of a median

\[m_a=\frac12\sqrt{2b^2+2c^2-a^2}\]
9–11 grade

Length of an angle bisector

\[l_a=\frac{2bc\cos\frac\alpha2}{b+c}\]
9–11 grade

Angle bisector theorem

\[\frac{BD}{DC}=\frac{AB}{AC}\]
8–11 grade