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Toolbox · Triangle calculator

Triangle calculator

Three sides, two sides with included angle, two angles with one side, or three coordinates - get all sides, angles, perimeter, area and circles

Formula

\cos A=\frac{b^2+c^2-a^2}{2bc}

Heron's formula S = sqrt[p(p-a)(p-b)(p-c)], law of cosines for angles

Enter sides a, b, c; any two must add up to more than the third.

abc

Presets

Tap an edge, angle or region to see how it is computed

Sides

Angles

Perimeter and circles

RelationS6

S=\sqrt{s(s-a)(s-b)(s-c)}=6

Area S6

Perimeter P12

Inradius r1

Circumradius R2.5

a3b4c5∠A36.87∠B53.13∠C90P12S6r1R2.5

Steps

9 / 9

1Given
$a=3,\ b=4,\ c=5$
List what is known, then work out the rest step by step
2Cosine rule
$\cos A=\frac{b^2+c^2-a^2}{2bc}=\frac{16+25-9}{2\times4\times5}=0.8$
cos A = (32) ÷ (40) = 0.8
3Cosine rule
$\cos B=\frac{a^2+c^2-b^2}{2ac}=0.6\Rightarrow B=53.13^\circ$
cos B = (18) ÷ (30) = 0.6
4Angle sum
$C=180^\circ-36.87^\circ-53.13^\circ=90^\circ$
The angles of a triangle add up to 180°, so the third is the remainder
5Perimeter
$P=a+b+c=3+4+5=12$
Add the three sides: 3 + 4 + 5 = 12
6Heron's formula
$s=\frac{P}{2}=6,\quad S=\sqrt{s(s-a)(s-b)(s-c)}=6$
Heron's formula: semiperimeter s = 6, area = √(s(s−a)(s−b)(s−c))
7Incircle
$r=\frac{S}{s}=\frac{6}{6}=1$
Incircle radius = area ÷ semiperimeter = 1
8Circumcircle
$R=\frac{abc}{4S}=\frac{3\times4\times5}{4\times6}=2.5$
Circumcircle radius = abc ÷ (4S) = 2.5
9Result
$a=3,\ b=4,\ c=5,\ A=36.87^\circ,\ B=53.13^\circ,\ C=90^\circ$
Area 6, perimeter 12, incircle 1, circumcircle 2.5

Concept

A triangle is the simplest polygon: its interior angles add up to 180 degrees and any two sides exceed the third.

Area = base x height / 2; use Heron for three sides, or ab sinC / 2 for two sides and the included angle.

Going further

Where Heron comes from: Heron writes the area symmetrically in the three sides: 16S^2 = 2a^2b^2 + 2b^2c^2 + 2c^2a^2 - a^4 - b^4 - c^4.
Sine and cosine rules: The sine rule links sides to opposite angles; the cosine rule generalises Pythagoras (it reduces to c^2 = a^2 + b^2 at 90 degrees).
Incircle and circumcircle: r = S/p with p the semiperimeter, and R = abc/(4S). For an equilateral triangle R = 2r, the largest possible ratio.

Common mistakes

The sides must satisfy the triangle inequality: 1, 2, 3 cannot form a triangle.
arccos from the cosine rule already returns 0-180 degrees, no need to test for obtuse.
Two sides plus a non-included angle (SSA) can give two triangles, so this tool asks for the included angle.

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