Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Obtuse scalene triangle.

Sides: a = 3.02114111139   b = 3.45503189872   c = 5.88

Area: T = 3.96435522727
Perimeter: p = 12.35217301011
Semiperimeter: s = 6.17658650505

Angle ∠ A = α = 23° = 0.4011425728 rad
Angle ∠ B = β = 26.5° = 26°30' = 0.46325122518 rad
Angle ∠ C = γ = 130.5° = 130°30' = 2.27876546739 rad

Height: ha = 2.62436431411
Height: hb = 2.29774990355
Height: hc = 1.34881470315

Median: ma = 4.57879164777
Median: mb = 4.34545928786
Median: mc = 1.36986537605

Inradius: r = 0.64217809068
Circumradius: R = 3.86663557797

Vertex coordinates: A[5.88; 0] B[0; 0] C[2.70439646263; 1.34881470315]
Centroid: CG[2.86113215421; 0.44993823438]
Coordinates of the circumscribed circle: U[2.94; -2.51109972153]
Coordinates of the inscribed circle: I[2.72655460633; 0.64217809068]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157° = 0.4011425728 rad
∠ B' = β' = 153.5° = 153°30' = 0.46325122518 rad
∠ C' = γ' = 49.5° = 49°30' = 2.27876546739 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 23° ; ; beta = 26° 30' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 23° - 26° 30' = 130° 30' ; ;

2. By using the law of sines, we calculate unknown side a

c = 5.88 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 5.88 * fraction{ sin 23° }{ sin 130° 30' } = 3.02 ; ;

3. By using the law of sines, we calculate last unknown side b

 fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 5.88 * fraction{ sin 26° 30' }{ sin 130° 30' } = 3.45 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 3.02 ; ; b = 3.45 ; ; c = 5.88 ; ;

4. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 3.02+3.45+5.88 = 12.35 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.35 }{ 2 } = 6.18 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.18 * (6.18-3.02)(6.18-3.45)(6.18-5.88) } ; ; T = sqrt{ 15.71 } = 3.96 ; ;

7. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.96 }{ 3.02 } = 2.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.96 }{ 3.45 } = 2.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.96 }{ 5.88 } = 1.35 ; ;

8. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 3.45**2+5.88**2-3.02**2 }{ 2 * 3.45 * 5.88 } ) = 23° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.02**2+5.88**2-3.45**2 }{ 2 * 3.02 * 5.88 } ) = 26° 30' ; ;
 gamma = 180° - alpha - beta = 180° - 23° - 26° 30' = 130° 30' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.96 }{ 6.18 } = 0.64 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.02 }{ 2 * sin 23° } = 3.87 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.45**2+2 * 5.88**2 - 3.02**2 } }{ 2 } = 4.578 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.88**2+2 * 3.02**2 - 3.45**2 } }{ 2 } = 4.345 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.45**2+2 * 3.02**2 - 5.88**2 } }{ 2 } = 1.369 ; ;
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