Triangle Calculator - Step-by-Step Solutions

Choose up to three elements of a triangle that you know, enter their values and choose what to find. The calculator solves the triangle step by step, with the formulas it uses, and draws it to scale. Under the drawing you can switch on each median, altitude and angle bisector, the inscribed circle, the circumscribed circle and the Euler line.

Examples: two sides and the angle between them an angle from the three sides a side and two angles two sides and an angle opposite one of them (two triangles) the area from the three medians the area from the three altitudes two sides and the radius of the inscribed circle a side, the altitude and the median to it the area, the perimeter and an angle the distance between the centers of the two circles an angle, the distance from the centroid to the orthocenter and the angle $AGH$

How to use the calculator

In each row "Given" choose an element from the list and type its value. If you know fewer than three elements, leave a row at "not given".
The sketch next to the form shows each element you give, with its value, and marks what is asked with a question mark. It is not to scale: the triangle is drawn to scale in the answer.
A value can be a whole number, a decimal, a fraction such as 7/2 or a square root such as sqrt(3) or 2*sqrt(3). Angles are in degrees.
In the row "Find" choose the element you need, or "everything" to get all sides and angles, the perimeter and the area.
For a distance between two points or an angle by three points, choose the points from the lists that appear. In an angle the vertex is the point in the middle: $\angle AGH$ is the angle at $G$.
The address of the page changes with the problem, so you can copy it and send the solution to someone.

Notation

In triangle $ABC$ the side ${a = BC}$ is opposite the vertex $A$, ${b = AC}$ is opposite $B$ and ${c = AB}$ is opposite $C$. The angles at $A$, $B$ and $C$ are $\alpha$, $\beta$ and $\gamma$.

Symbol	Meaning
$h_a$, $m_a$, $l_a$	the altitude, the median and the angle bisector from the vertex $A$ to the side $a$
$S$, $P$, $p$	the area, the perimeter and the semiperimeter ${p = \frac{a + b + c}{2}}$
$R$, $r$	the radii of the circumscribed and of the inscribed circle
$r_a$	the radius of the escribed circle that touches the side $a$
$G$, $H$, $O$, $I$	the centroid (where the medians meet), the orthocenter (where the altitudes meet), the circumcenter and the incenter
$M_a$, $H_a$, $L_a$, $T_a$	on the side $a$: its midpoint, the foot of the altitude from $A$, the foot of the bisector from $A$ and the point where the inscribed circle touches it

How the calculator solves a triangle

A triangle has three degrees of freedom, so three independent elements, at least one of them a length, determine it - sometimes in two ways. The calculator works like a student with a formula sheet.

Formulas, one after another. From the given elements it looks for a formula in which everything but one element is known, finds that element, and repeats until it reaches what you asked for. Three sides give an angle by the law of cosines; two angles give the third; a side and the altitude to it give the area.
An unknown and an equation. When no formula gives a new element directly - for example two sides and the radius of the inscribed circle - the calculator calls one element $x$, writes the other elements through $x$, and the given element that is left over becomes an equation for $x$. The equation is solved numerically.
Coordinates. Elements such as the angle $AGH$ have no ready formula. The calculator puts the triangle in a coordinate system, ${A(0, 0)}$, ${B(c, 0)}$, ${C(b\cos\alpha, b\sin\alpha)}$, finds the points and measures the distance or the angle there.

Exact values such as ${\sqrt{39}}$ or ${\frac{35\sqrt3}{4}}$ are kept exact where possible, and every answer is checked against the triangle that is drawn.

Formulas for solving triangles

Sides and angles

Name	Formula
Sum of the angles	${\alpha + \beta + \gamma = 180^\circ}$
Law of sines	${\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} = 2R}$
Law of cosines	${a^2 = b^2 + c^2 - 2bc\cos\alpha}$
An angle from the sides	${\cos\alpha = \frac{b^2 + c^2 - a^2}{2bc}}$

Area

Known	Formula
A side and its altitude	${S = \frac{a h_a}{2}}$
Two sides and the angle between them	${S = \frac{bc\sin\alpha}{2}}$
Three sides (Heron's formula)	${S = \sqrt{p(p - a)(p - b)(p - c)}}$
The inscribed circle	${S = pr}$
The circumscribed circle	${S = \frac{abc}{4R}}$

Altitudes, medians and bisectors

Element	Formula
Altitude	${h_a = \frac{2S}{a} = b\sin\gamma = c\sin\beta}$
Median	${m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}}$
Angle bisector	${l_a = \frac{2bc\cos\frac{\alpha}{2}}{b + c}}$
Where the bisector meets the side	${BL_a : L_aC = c : b}$

Circles and centers

Element	Formula
Radius of the inscribed circle	${r = \frac{S}{p} = (p - a)\tan\frac{\alpha}{2}}$
Radius of the circumscribed circle	${R = \frac{a}{2\sin\alpha} = \frac{abc}{4S}}$
Radius of an escribed circle	${r_a = \frac{S}{p - a}}$
Distance between the centers of the two circles (Euler)	${OI^2 = R^2 - 2Rr}$
Centroid	${AG = \frac{2}{3}m_a}$
Orthocenter	${AH = 2R\,\|\cos\alpha\|}$, ${OH^2 = 9R^2 - (a^2 + b^2 + c^2)}$
Euler line	$O$, $G$ and $H$ are on one line and ${GH = 2\,OG}$

When is a triangle determined?

Three sides determine a triangle when each is shorter than the other two together.
Two sides and the angle between them, or a side and two angles, always determine one triangle.
Two sides and an angle opposite one of them may give two triangles, one or none: two angles between $0^\circ$ and $180^\circ$ have the same sine. The calculator shows every triangle that fits.
Three angles give the shape only. At least one length or the area is needed for the size.
Elements that depend on each other, such as a side, the opposite angle and $R$ (because ${a = 2R\sin\alpha}$), count as two. The calculator then tells you that the triangle is not determined, or finds what the data do determine.

Other resources

Triangles Law of sines and law of cosines Sine rule problems Cosine rule problems Area formulas Altitude of a triangle Medians of a triangle Problem solver