Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 10.4066141637   b = 6.46109434604   c = 14.6

Area: T = 29.68218252948
Perimeter: p = 31.46770850974
Semiperimeter: s = 15.73435425487

Angle ∠ A = α = 39° = 0.68106784083 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 118° = 2.05994885174 rad

Height: ha = 5.70546744759
Height: hb = 9.18880777093
Height: hc = 4.0666003465

Median: ma = 10.01989794519
Median: mb = 12.25991983541
Median: mc = 4.66110929066

Inradius: r = 1.88765316061
Circumradius: R = 8.268776137

Vertex coordinates: A[14.6; 0] B[0; 0] C[9.57989038826; 4.0666003465]
Centroid: CG[8.06596346275; 1.35553344883]
Coordinates of the circumscribed circle: U[7.3; -3.88114788511]
Coordinates of the inscribed circle: I[9.27325990883; 1.88765316061]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141° = 0.68106784083 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 62° = 2.05994885174 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 39° ; ; beta = 23° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 39° - 23° = 118° ; ;

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

c = 14.6 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 14.6 * fraction{ sin(39° ) }{ sin (118° ) } = 10.41 ; ;

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 = 14.6 * fraction{ sin(23° ) }{ sin (118° ) } = 6.46 ; ;


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 = 10.41 ; ; b = 6.46 ; ; c = 14.6 ; ;

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

p = a+b+c = 10.41+6.46+14.6 = 31.47 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.47 }{ 2 } = 15.73 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.73 * (15.73-10.41)(15.73-6.46)(15.73-14.6) } ; ; T = sqrt{ 881.01 } = 29.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.68 }{ 10.41 } = 5.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.68 }{ 6.46 } = 9.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.68 }{ 14.6 } = 4.07 ; ;

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{ 10.41**2-6.46**2-14.6**2 }{ 2 * 6.46 * 14.6 } ) = 39° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.46**2-10.41**2-14.6**2 }{ 2 * 10.41 * 14.6 } ) = 23° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.6**2-10.41**2-6.46**2 }{ 2 * 6.46 * 10.41 } ) = 118° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.68 }{ 15.73 } = 1.89 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.41 }{ 2 * sin 39° } = 8.27 ; ;




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