Triangle Solver
Every side, every angle, the area and the two radii of any triangle, from whichever three measurements you have.
Results
Side · its length · angle · its size
a 3 A 36.869898
b 4 B 53.130102
c 5 C 90
Side a
3.0000
m
Side b
4.0000
m
Side c
5.0000
m
Angle A, opposite side a
36.8699
°
Angle B, opposite side b
53.1301
°
Angle C, opposite side c
90.0000
°
Area
6.0000
m²
Perimeter
12.0000
m
How many triangles fit
Exactly one
By its angles
Right-angled
By its sides
Scalene — all three different
Inradius, the circle that fits inside
1.0000
m
Circumradius, the circle through the corners
2.5000
m
Height measured onto side a
4.0000
m
Semi-perimeter
6.0000
m
The three angles added up
180.0000
°
Angle B of the second triangle
—
°
Side c of the second triangle
—
m
Area of the second triangle
—
m²
What this tool does
Three measurements settle a triangle — but which three you have decides how it is solved. Three sides go through the law of cosines; two angles and a side go through the angle sum and then the law of sines. One combination is special: two sides and an angle that is not between them can fit two different triangles, and this page says so instead of picking one and staying quiet about the other.
Formula
SSS and SAS: law of cosines · ASA and AAS: A + B + C = 180° and then the law of sines · SSA: the law of sines, and it can give two triangles
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mode | What you know | — |
a | Side a | m |
b | Side b | m |
c | Side c | m |
alpha | Angle A, opposite side a | ° |
beta | Angle B, opposite side b | ° |
gamma | Angle C, opposite side c | ° |
TB | Side · its length · angle · its size | — |
SA | Side a | m |
SB | Side b | m |
SC | Side c | m |
AA | Angle A, opposite side a | ° |
AB | Angle B, opposite side b | ° |
AC | Angle C, opposite side c | ° |
AR | Area | m² |
PE | Perimeter | m |
NS | How many triangles fit | — |
KD | By its angles | — |
KS | By its sides | — |
IR | Inradius, the circle that fits inside | m |
CR | Circumradius, the circle through the corners | m |
HA | Height measured onto side a | m |
SP | Semi-perimeter | m |
SM | The three angles added up | ° |
B2 | Angle B of the second triangle | ° |
C2 | Side c of the second triangle | m |
R2 | Area of the second triangle | m² |
Worked example
- What you knowsss
- Side a3 m
- Side b4 m
- Side c5 m
- Angle A, opposite side a0 °
- Angle B, opposite side b0 °
- Angle C, opposite side c0 °
- Side · its length · angle · its sizea 3 A 36.869898 b 4 B 53.130102 c 5 C 90
- Side a3.0000 m
- Side b4.0000 m
- Side c5.0000 m
- Angle A, opposite side a36.8699 °
- Angle B, opposite side b53.1301 °
- Angle C, opposite side c90.0000 °
- Area6.0000 m²
- Perimeter12.0000 m
- How many triangles fitExactly one
- By its anglesRight-angled
- By its sidesScalene — all three different
- Inradius, the circle that fits inside1.0000 m
- Circumradius, the circle through the corners2.5000 m
- Height measured onto side a4.0000 m
- Semi-perimeter6.0000 m
- The three angles added up180.0000 °
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.