Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 34   b = 23.79899482088   c = 32.22224317793

Area: T = 366.5377260352
Perimeter: p = 90.01223799881
Semiperimeter: s = 45.00661899941

Angle ∠ A = α = 73° = 1.2744090354 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 65° = 1.13444640138 rad

Height: ha = 21.56110153148
Height: hb = 30.81444647592
Height: hc = 22.75504406162

Median: ma = 22.65222266626
Median: mb = 30.91436239536
Median: mc = 24.52436526734

Inradius: r = 8.144415218
Circumradius: R = 17.77767598603

Vertex coordinates: A[32.22224317793; 0] B[0; 0] C[25.26769240662; 22.75504406162]
Centroid: CG[19.16331186152; 7.58334802054]
Coordinates of the circumscribed circle: U[16.11112158897; 7.51327833515]
Coordinates of the inscribed circle: I[21.21662417852; 8.144415218]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107° = 1.2744090354 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 115° = 1.13444640138 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 73° ; ; beta = 42° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 73° - 42° = 65° ; ;

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

a = 34 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 34 * fraction{ sin(42° ) }{ sin (73° ) } = 23.79 ; ;

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

 fraction{ c }{ a } = fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = a * fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = 34 * fraction{ sin(65° ) }{ sin (73° ) } = 32.22 ; ;


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 = 34 ; ; b = 23.79 ; ; c = 32.22 ; ;

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

p = a+b+c = 34+23.79+32.22 = 90.01 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.01 }{ 2 } = 45.01 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.01 * (45.01-34)(45.01-23.79)(45.01-32.22) } ; ; T = sqrt{ 134349.56 } = 366.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 366.54 }{ 34 } = 21.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 366.54 }{ 23.79 } = 30.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 366.54 }{ 32.22 } = 22.75 ; ;

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{ 34**2-23.79**2-32.22**2 }{ 2 * 23.79 * 32.22 } ) = 73° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23.79**2-34**2-32.22**2 }{ 2 * 34 * 32.22 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 32.22**2-34**2-23.79**2 }{ 2 * 23.79 * 34 } ) = 65° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 366.54 }{ 45.01 } = 8.14 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 34 }{ 2 * sin 73° } = 17.78 ; ;




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