Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 22.01217524755   b = 19.99769033079   c = 35.651104

Area: T = 197.7255419759
Perimeter: p = 77.66596957835
Semiperimeter: s = 38.83298478917

Angle ∠ A = α = 33.69° = 33°41'24″ = 0.5888001425 rad
Angle ∠ B = β = 30.26° = 30°15'36″ = 0.52881366317 rad
Angle ∠ C = γ = 116.05° = 116°3' = 2.02554545969 rad

Height: ha = 17.96554409597
Height: hb = 19.77656039237
Height: hc = 11.09222665796

Median: ma = 26.72765240114
Median: mb = 27.88988492876
Median: mc = 11.15656053715

Inradius: r = 5.09220987461
Circumradius: R = 19.84111606291

Vertex coordinates: A[35.651104; 0] B[0; 0] C[19.0132597644; 11.09222665796]
Centroid: CG[18.2211212548; 3.69774221932]
Coordinates of the circumscribed circle: U[17.826552; -8.71333513552]
Coordinates of the inscribed circle: I[18.83329445838; 5.09220987461]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.31° = 146°18'36″ = 0.5888001425 rad
∠ B' = β' = 149.74° = 149°44'24″ = 0.52881366317 rad
∠ C' = γ' = 63.95° = 63°57' = 2.02554545969 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 33° 41'24" ; ; beta = 30° 15'36" ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 33° 41'24" - 30° 15'36" = 116° 3' ; ;

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

c = 35.65 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 35.65 * fraction{ sin(33° 41'24") }{ sin (116° 3') } = 22.01 ; ;

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 = 35.65 * fraction{ sin(30° 15'36") }{ sin (116° 3') } = 20 ; ;


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 = 22.01 ; ; b = 20 ; ; c = 35.65 ; ;

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

p = a+b+c = 22.01+20+35.65 = 77.66 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 77.66 }{ 2 } = 38.83 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.83 * (38.83-22.01)(38.83-20)(38.83-35.65) } ; ; T = sqrt{ 39095.34 } = 197.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 197.73 }{ 22.01 } = 17.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 197.73 }{ 20 } = 19.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 197.73 }{ 35.65 } = 11.09 ; ;

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{ 22.01**2-20**2-35.65**2 }{ 2 * 20 * 35.65 } ) = 33° 41'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-22.01**2-35.65**2 }{ 2 * 22.01 * 35.65 } ) = 30° 15'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 35.65**2-22.01**2-20**2 }{ 2 * 20 * 22.01 } ) = 116° 3' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 197.73 }{ 38.83 } = 5.09 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22.01 }{ 2 * sin 33° 41'24" } = 19.84 ; ;




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