Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 26.8   b = 13.59765012766   c = 22.14332512416

Area: T = 150.1550119194
Perimeter: p = 62.54397525182
Semiperimeter: s = 31.27698762591

Angle ∠ A = α = 94.1° = 94°6' = 1.64223548261 rad
Angle ∠ B = β = 30.4° = 30°24' = 0.53105800926 rad
Angle ∠ C = γ = 55.5° = 55°30' = 0.96986577349 rad

Height: ha = 11.20552327757
Height: hb = 22.08765818551
Height: hc = 13.56217048784

Median: ma = 12.5711165867
Median: mb = 23.62334116087
Median: mc = 18.13875723181

Inradius: r = 4.80217497079
Circumradius: R = 13.43443814984

Vertex coordinates: A[22.14332512416; 0] B[0; 0] C[23.11553663348; 13.56217048784]
Centroid: CG[15.08662058588; 4.52105682928]
Coordinates of the circumscribed circle: U[11.07216256208; 7.60993174699]
Coordinates of the inscribed circle: I[17.67333749825; 4.80217497079]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 85.9° = 85°54' = 1.64223548261 rad
∠ B' = β' = 149.6° = 149°36' = 0.53105800926 rad
∠ C' = γ' = 124.5° = 124°30' = 0.96986577349 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 94° 6' ; ; beta = 30° 24' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 94° 6' - 30° 24' = 55° 30' ; ;

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

a = 26.8 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 26.8 * fraction{ sin(30° 24') }{ sin (94° 6') } = 13.6 ; ;

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 = 26.8 * fraction{ sin(55° 30') }{ sin (94° 6') } = 22.14 ; ;


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 = 26.8 ; ; b = 13.6 ; ; c = 22.14 ; ;

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

p = a+b+c = 26.8+13.6+22.14 = 62.54 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62.54 }{ 2 } = 31.27 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.27 * (31.27-26.8)(31.27-13.6)(31.27-22.14) } ; ; T = sqrt{ 22545.06 } = 150.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 150.15 }{ 26.8 } = 11.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 150.15 }{ 13.6 } = 22.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 150.15 }{ 22.14 } = 13.56 ; ;

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{ 26.8**2-13.6**2-22.14**2 }{ 2 * 13.6 * 22.14 } ) = 94° 6' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13.6**2-26.8**2-22.14**2 }{ 2 * 26.8 * 22.14 } ) = 30° 24' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22.14**2-26.8**2-13.6**2 }{ 2 * 13.6 * 26.8 } ) = 55° 30' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 150.15 }{ 31.27 } = 4.8 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 26.8 }{ 2 * sin 94° 6' } = 13.43 ; ;




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