Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 13.03655662089   b = 44.1244353482   c = 54

Area: T = 206.8777066468
Perimeter: p = 111.1659919691
Semiperimeter: s = 55.58799598454

Angle ∠ A = α = 10° = 0.17545329252 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 134° = 2.33987411977 rad

Height: ha = 31.74404036238
Height: hb = 9.3777001594
Height: hc = 7.66221135729

Median: ma = 48.87773750165
Median: mb = 32.54995900073
Median: mc = 18.15105448484

Inradius: r = 3.72221521398
Circumradius: R = 37.53444169575

Vertex coordinates: A[54; 0] B[0; 0] C[10.54659945943; 7.66221135729]
Centroid: CG[21.51553315314; 2.55440378576]
Coordinates of the circumscribed circle: U[27; -26.07435969198]
Coordinates of the inscribed circle: I[11.45656063635; 3.72221521398]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170° = 0.17545329252 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 46° = 2.33987411977 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 10° ; ; beta = 36° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 10° - 36° = 134° ; ;

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

c = 54 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 54 * fraction{ sin(10° ) }{ sin (134° ) } = 13.04 ; ;

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 = 54 * fraction{ sin(36° ) }{ sin (134° ) } = 44.12 ; ;


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 = 13.04 ; ; b = 44.12 ; ; c = 54 ; ;

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

p = a+b+c = 13.04+44.12+54 = 111.16 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 111.16 }{ 2 } = 55.58 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 55.58 * (55.58-13.04)(55.58-44.12)(55.58-54) } ; ; T = sqrt{ 42798.12 } = 206.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 206.88 }{ 13.04 } = 31.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 206.88 }{ 44.12 } = 9.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 206.88 }{ 54 } = 7.66 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13.04**2-44.12**2-54**2 }{ 2 * 44.12 * 54 } ) = 10° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 44.12**2-13.04**2-54**2 }{ 2 * 13.04 * 54 } ) = 36° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 54**2-13.04**2-44.12**2 }{ 2 * 44.12 * 13.04 } ) = 134° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 206.88 }{ 55.58 } = 3.72 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.04 }{ 2 * sin 10° } = 37.53 ; ;




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