Triangle calculator ASA

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


Acute scalene triangle.

Sides: a = 16.22994498946   b = 17.98882054232   c = 9.3

Area: T = 75.18797675116
Perimeter: p = 43.51876553178
Semiperimeter: s = 21.75988276589

Angle ∠ A = α = 64° = 1.11770107213 rad
Angle ∠ B = β = 85° = 1.48435298642 rad
Angle ∠ C = γ = 31° = 0.54110520681 rad

Height: ha = 9.26546106923
Height: hb = 8.35987846306
Height: hc = 16.1687691938

Median: ma = 11.79876271429
Median: mb = 9.69878677221
Median: mc = 16.48882621618

Inradius: r = 3.45551387
Circumradius: R = 9.02884587228

Vertex coordinates: A[9.3; 0] B[0; 0] C[1.414448976; 16.1687691938]
Centroid: CG[3.57114965867; 5.3899230646]
Coordinates of the circumscribed circle: U[4.65; 7.73988995929]
Coordinates of the inscribed circle: I[3.77106222357; 3.45551387]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116° = 1.11770107213 rad
∠ B' = β' = 95° = 1.48435298642 rad
∠ C' = γ' = 149° = 0.54110520681 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 64° ; ; beta = 85° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 64° - 85° = 31° ; ;

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

c = 9.3 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 9.3 * fraction{ sin(64° ) }{ sin (31° ) } = 16.23 ; ;

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 = 9.3 * fraction{ sin(85° ) }{ sin (31° ) } = 17.99 ; ;


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 = 16.23 ; ; b = 17.99 ; ; c = 9.3 ; ;

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

p = a+b+c = 16.23+17.99+9.3 = 43.52 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43.52 }{ 2 } = 21.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.76 * (21.76-16.23)(21.76-17.99)(21.76-9.3) } ; ; T = sqrt{ 5652 } = 75.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 75.18 }{ 16.23 } = 9.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 75.18 }{ 17.99 } = 8.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 75.18 }{ 9.3 } = 16.17 ; ;

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{ 16.23**2-17.99**2-9.3**2 }{ 2 * 17.99 * 9.3 } ) = 64° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17.99**2-16.23**2-9.3**2 }{ 2 * 16.23 * 9.3 } ) = 85° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.3**2-16.23**2-17.99**2 }{ 2 * 17.99 * 16.23 } ) = 31° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 75.18 }{ 21.76 } = 3.46 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.23 }{ 2 * sin 64° } = 9.03 ; ;




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