Triangle calculator AAS

Please enter two angles and one opposite side
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°


Obtuse scalene triangle.

Sides: a = 40   b = 15.75553872771   c = 32.21986717611

Area: T = 241.3876537456
Perimeter: p = 87.97440590382
Semiperimeter: s = 43.98770295191

Angle ∠ A = α = 108° = 1.88549555922 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 12.06993268728
Height: hb = 30.64217777248
Height: hc = 14.98442637366

Median: ma = 15.5932867573
Median: mb = 35.45436789059
Median: mc = 25.78799420405

Inradius: r = 5.48876753465
Circumradius: R = 21.02992444848

Vertex coordinates: A[32.21986717611; 0] B[0; 0] C[37.08773541827; 14.98442637366]
Centroid: CG[23.10220086479; 4.99547545789]
Coordinates of the circumscribed circle: U[16.10993358805; 13.51773377959]
Coordinates of the inscribed circle: I[28.2321642242; 5.48876753465]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72° = 1.88549555922 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 108° ; ; beta = 22° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 108° - 22° = 50° ; ;

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

a = 40 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 40 * fraction{ sin 22° }{ sin 108° } = 15.76 ; ;

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 = 40 * fraction{ sin 50° }{ sin 108° } = 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 = 40 ; ; b = 15.76 ; ; c = 32.22 ; ;

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

p = a+b+c = 40+15.76+32.22 = 87.97 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 87.97 }{ 2 } = 43.99 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.99 * (43.99-40)(43.99-15.76)(43.99-32.22) } ; ; T = sqrt{ 58267.46 } = 241.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 241.39 }{ 40 } = 12.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 241.39 }{ 15.76 } = 30.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 241.39 }{ 32.22 } = 14.98 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 15.76**2+32.22**2-40**2 }{ 2 * 15.76 * 32.22 } ) = 108° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+32.22**2-15.76**2 }{ 2 * 40 * 32.22 } ) = 22° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 40**2+15.76**2-32.22**2 }{ 2 * 40 * 15.76 } ) = 50° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 241.39 }{ 43.99 } = 5.49 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 40 }{ 2 * sin 108° } = 21.03 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.76**2+2 * 32.22**2 - 40**2 } }{ 2 } = 15.593 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.22**2+2 * 40**2 - 15.76**2 } }{ 2 } = 35.454 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.76**2+2 * 40**2 - 32.22**2 } }{ 2 } = 25.78 ; ;
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