MENUGeometry
Triangle formulas
In triangle $ABC$ the sides are ${a=BC},$ ${b=CA},$ ${c=AB},$ and the angles at the vertices are $\alpha,$ $\beta$ and $\gamma.$ $R$ and $r$ are the radii of the circumscribed and inscribed circles, and ${p=\frac{a+b+c}{2}}$ is the semiperimeter.
Relations in a triangle
| Rule | Formula |
|---|---|
| Angle sum | ${\alpha+\beta+\gamma=180^\circ}$ |
| Triangle inequality | ${|b-c|\lt a\lt b+c}$ |
| Law of cosines | ${a^2=b^2+c^2-2bc\cos\alpha}$ ${b^2=a^2+c^2-2ac\cos\beta}$ ${c^2=a^2+b^2-2ab\cos\gamma}$ |
| Law of sines | ${\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R}$ |
Median
| Rule | Formula |
|---|---|
| Length of the medians | ${m_a=\frac12\sqrt{2b^2+2c^2-a^2}}$ ${m_b=\frac12\sqrt{2a^2+2c^2-b^2}}$ ${m_c=\frac12\sqrt{2a^2+2b^2-c^2}}$ |
| Centroid | The medians meet at one point $G,$ which divides each of them in the ratio ${2:1}$ from the vertex: ${AG:GA_1=2:1}$ |
More in the lesson Median of a triangle.
Angle bisector
| Rule | Formula |
|---|---|
| Angle bisector theorem | ${\frac{a}{b}=\frac{m}{n}}$ – the bisector divides the opposite side into parts proportional to the adjacent sides |
| Length of the bisector | ${l^2=ab-mn}$ |
Right triangle
Here ${\gamma=90^\circ},$ $a$ and $b$ are the legs, $c$ is the hypotenuse, ${h=CH}$ is the altitude to the hypotenuse, and ${n=HB}$ and ${m=AH}$ are the projections of the legs on the hypotenuse.
| Rule | Formula |
|---|---|
| Pythagorean theorem | ${c^2=a^2+b^2}$ |
| Area | ${S=\frac{ab}{2}=\frac{ch}{2}}$ |
| Leg and its projection | ${a^2=c\cdot n},$ ${b^2=c\cdot m}$ |
| Altitude to the hypotenuse | ${h^2=n\cdot m},$ ${h=\frac{ab}{c}}$ |
| Trigonometric functions | ${\sin\alpha=\frac{a}{c}},$ ${\cos\alpha=\frac{b}{c}},$ ${\tan\alpha=\frac{a}{b}},$ ${\cot\alpha=\frac{b}{a}}$ |
| Radii | ${R=\frac{c}{2}},$ ${r=\frac{a+b-c}{2}}$ |
More in the lesson Triangles.
Area formulas
Area of a triangle
| Rule | Formula |
|---|---|
| A side and the altitude to it | ${S=\frac{c\cdot h_c}{2}}$ |
| Two sides and the included angle | ${S=\frac12ab\sin\gamma}$ |
| Heron's formula | ${S}={\sqrt{p(p-a)(p-b)(p-c)}}$ |
| With the inradius | ${S=p\cdot r}$ |
| With the circumradius | ${S=\frac{abc}{4R}}$ |
Area of a parallelogram
| Rule | Formula |
|---|---|
| A side and the altitude to it | ${S=a\cdot h_a=b\cdot h_b}$ – in the figure ${S=AB\cdot DE=BC\cdot DF}$ |
| Two sides and the angle between them | ${S=ab\sin\alpha}$ |
| Rectangle and square | ${S=ab},$ ${S=a^2}$ |
Area of a quadrilateral
| Rule | Formula |
|---|---|
| Convex quadrilateral (also a parallelogram) | ${S=\frac12d_1d_2\sin\varphi},$ $\varphi$ – the angle between the diagonals |
| Rhombus, square and any quadrilateral with perpendicular diagonals | ${S=\frac{d_1d_2}{2}}$ |
| Trapezoid with bases $a,$ $b$ and height $h$ | ${S=\frac{a+b}{2}\cdot h}$ |
Area of a circumscribed polygon
| Rule | Formula |
|---|---|
| Polygon circumscribed about a circle of radius $r$ | ${S=p\cdot r=\frac{P\cdot r}{2}},$ $P$ – the perimeter, $p$ – the semiperimeter |
Circle
| Rule | Formula |
|---|---|
| Circumference | ${C=2\pi r}$ |
| Area of a circle | ${S=\pi r^2}$ |
| Arc length for a central angle of $\alpha^\circ$ | ${\ell=\frac{\pi r\alpha}{180}}$ |
| Area of a sector with a central angle of $\alpha^\circ$ | ${S=\frac{\pi r^2\alpha}{360}}$ |
Practice
Interactive exercises with answer checking:
