Formulas and reference

Adjacent supplementary angles

\[\alpha+\beta=180^\circ\]

7–9 grade

Vertical angles

\[\alpha=\beta\]

7–9 grade

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

Pythagorean theorem

NMT

\[a^2+b^2=c^2\]

8–11 grade

Cosine of an acute angle

NMT

\[\cos\alpha=\frac bc\]

8–11 grade

Sine of an acute angle

NMT

\[\sin\alpha=\frac ac\]

8–11 grade

Tangent of an acute angle

NMT

\[\tg\alpha=\frac ab\]

8–11 grade

Altitude to the hypotenuse

\[h=\frac{ab}{c}\]

8–11 grade

Altitude through the leg projections

\[h^2=pq\]

8–11 grade

Leg from its projection on the hypotenuse

\[a^2=cp\]

8–11 grade

Second leg from its projection

\[b^2=cq\]

8–11 grade

Circumradius of a right triangle

\[R=\frac c2\]

8–11 grade

Altitude of an equilateral triangle

\[h=\frac{a\sqrt3}{2}\]

8–11 grade

Area of an equilateral triangle

\[S=\frac{a^2\sqrt3}{4}\]

8–11 grade

Circumradius of an equilateral triangle

\[R=\frac{a\sqrt3}{3}\]

8–11 grade

Inradius of an equilateral triangle

\[r=\frac{a\sqrt3}{6}\]

8–11 grade

Area of a parallelogram

NMT

\[S=ah_a\]

8–11 grade

Parallelogram area from two sides and an angle

NMT

\[S=ab\sin\gamma\]

9–11 grade

Sum of squares of parallelogram diagonals

\[d_1^2+d_2^2=2(a^2+b^2)\]

8–11 grade

Area of a rectangle

NMT

\[S=ab\]

5–11 grade

Diagonal of a rectangle

\[d=\sqrt{a^2+b^2}\]

8–11 grade

Rhombus area from diagonals

NMT

\[S=\frac12d_1d_2\]

8–11 grade

Rhombus area from a side and angle

\[S=a^2\sin\alpha\]

9–11 grade

Area of a square

\[S=a^2\]

5–11 grade

Diagonal of a square

\[d=a\sqrt2\]

8–11 grade

Midline of a trapezoid

\[m=\frac{a+b}{2}\]

8–11 grade

Area of a trapezoid

NMT

\[S=\frac{a+b}{2}\,h\]

8–11 grade

Sum of interior angles of an n-gon

\[\sum\alpha=(n-2)180^\circ\]

8–11 grade

Central angle of a regular n-gon

\[\varphi=\frac{360^\circ}{n}\]

8–11 grade

Area of a regular polygon

\[S=\frac12Pr\]

8–11 grade

Circumference

NMT

\[L=2\pi R\]

7–11 grade

Arc length

\[l=\frac{\alpha}{360^\circ}\,2\pi R\]

8–11 grade

Chord length

\[AB=2R\sin\frac{\alpha}{2}\]

9–11 grade

Inscribed angle

\[\angle ACB=\frac12\overset{\frown}{AB}\]

8–11 grade

Radius to the point of tangency

\[OT\perp t\]

7–11 grade

Intersecting chords theorem

\[PA\cdot PB=PC\cdot PD\]

8–11 grade

Tangent–secant theorem

\[PT^2=PA\cdot PB\]

8–11 grade

Area of a circle

NMT

\[S=\pi R^2\]

7–11 grade

Area of a sector

\[S_{\rm сектора}=\frac{\alpha}{360^\circ}\pi R^2\]

8–11 grade

Midpoint coordinates in the plane

\[x_0=\frac{x_1+x_2}{2}\]

\[y_0=\frac{y_1+y_2}{2}\]

9–11 grade

Distance between two points in the plane

\[AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

9–11 grade

Equation of a circle

NMT

\[(x-x_0)^2+(y-y_0)^2=R^2\]

9–11 grade

Equation of a line

\[Ax+By+C=0\]

Condition \(A^2+B^2\ne0\)

9–11 grade

Midpoint coordinates in space

NMT

\[x_0=\frac{x_1+x_2}{2}\]

\[y_0=\frac{y_1+y_2}{2}\]

\[z_0=\frac{z_1+z_2}{2}\]

10–11 grade

Distance between two points in space

NMT

\[AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]

10–11 grade

Coordinates of vector AB

NMT

\[\overrightarrow{AB}=(x_2-x_1;\,y_2-y_1;\,z_2-z_1)\]

10–11 grade

Vector length

\[|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2}\]

10–11 grade

Dot product in coordinates

NMT

\[\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3\]

9–11 grade

Dot product using the angle

NMT

\[\vec a\cdot\vec b=|\vec a||\vec b|\cos\varphi\]

9–11 grade

Perpendicular vectors

\[\vec a\perp\vec b\Longleftrightarrow\vec a\cdot\vec b=0\]

9–11 grade

Collinear vectors in the plane

\[a_1b_2-a_2b_1=0\]

9–11 grade

Homothety: lengths

\[l'=|k|l\]

9–11 grade

Homothety: areas

\[S'=k^2S\]

9–11 grade

Similar figures: corresponding lengths

\[\frac{a'}a=k\]

8–11 grade

Similar figures: areas

\[\frac{S'}S=k^2\]

8–11 grade

Volume of a right prism

NMT

\[V=S_{\rm осн}H\]

10–11 grade

Lateral surface area of a right prism

NMT

\[S_{\rm біч}=P_{\rm осн}H\]

10–11 grade

Total surface area of a prism

\[S_{\rm повн}=S_{\rm біч}+2S_{\rm осн}\]

10–11 grade

Volume of a rectangular cuboid

\[V=abc\]

10–11 grade

Diagonal of a rectangular cuboid

\[d=\sqrt{a^2+b^2+c^2}\]

10–11 grade

Volume of a cube

\[V=a^3\]

10–11 grade

Surface area of a cube

\[S_{\rm повн}=6a^2\]

10–11 grade

Diagonal of a cube

\[d=a\sqrt3\]

10–11 grade

Volume of a pyramid

NMT

\[V=\frac13S_{\rm осн}H\]

11–11 grade

Lateral surface area of a regular pyramid

NMT

\[S_{\rm біч}=\frac12P_{\rm осн}m\]

m — apothem.

11–11 grade

Total surface area of a pyramid

\[S_{\rm повн}=S_{\rm біч}+S_{\rm осн}\]

11–11 grade

Volume of a cylinder

NMT

\[V=\pi R^2H\]

11–11 grade

Lateral surface area of a cylinder

NMT

\[S_{\rm біч}=2\pi RH\]

11–11 grade

Total surface area of a cylinder

\[S_{\rm повн}=2\pi R(R+H)\]

11–11 grade

Slant height of a right cone

\[L=\sqrt{R^2+H^2}\]

11–11 grade

Volume of a cone

NMT

\[V=\frac13\pi R^2H\]

11–11 grade

Lateral surface area of a cone

NMT

\[S_{\rm біч}=\pi RL\]

11–11 grade

Total surface area of a cone

\[S_{\rm повн}=\pi R(R+L)\]

11–11 grade

Volume of a sphere

NMT

\[V=\frac43\pi R^3\]

11–11 grade

Surface area of a sphere

NMT

\[S=4\pi R^2\]

11–11 grade