Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Obtuse scalene triangle.

Sides: a = 32   b = 12.5   c = 22.23877192949

Area: T = 103.0087614982
Perimeter: p = 66.73877192949
Semiperimeter: s = 33.36988596474

Angle ∠ A = α = 132.1721555417° = 132°10'18″ = 2.30768288195 rad
Angle ∠ B = β = 16.82884445827° = 16°49'42″ = 0.2943711766 rad
Angle ∠ C = γ = 31° = 0.54110520681 rad

Height: ha = 6.43879759364
Height: hb = 16.48112183971
Height: hc = 9.2644224772

Median: ma = 8.33296506361
Median: mb = 26.83664598954
Median: mc = 21.5998517545

Inradius: r = 3.08769384231
Circumradius: R = 21.58884226606

Vertex coordinates: A[22.23877192949; 0] B[0; 0] C[30.63296284564; 9.2644224772]
Centroid: CG[17.62224492504; 3.0888074924]
Coordinates of the circumscribed circle: U[11.11988596474; 18.50548899784]
Coordinates of the inscribed circle: I[20.86988596474; 3.08769384231]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.82884445827° = 47°49'42″ = 2.30768288195 rad
∠ B' = β' = 163.1721555417° = 163°10'18″ = 0.2943711766 rad
∠ C' = γ' = 149° = 0.54110520681 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 32 ; ; b = 12.5 ; ; gamma = 31° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 32**2+12.5**2 - 2 * 32 * 12.5 * cos 31° } ; ; c = 22.24 ; ;


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 = 32 ; ; b = 12.5 ; ; c = 22.24 ; ;

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

p = a+b+c = 32+12.5+22.24 = 66.74 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66.74 }{ 2 } = 33.37 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.37 * (33.37-32)(33.37-12.5)(33.37-22.24) } ; ; T = sqrt{ 10610.57 } = 103.01 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.01 }{ 32 } = 6.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.01 }{ 12.5 } = 16.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.01 }{ 22.24 } = 9.26 ; ;

6. 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{ 12.5**2+22.24**2-32**2 }{ 2 * 12.5 * 22.24 } ) = 132° 10'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 32**2+22.24**2-12.5**2 }{ 2 * 32 * 22.24 } ) = 16° 49'42" ; ; gamma = 180° - alpha - beta = 180° - 132° 10'18" - 16° 49'42" = 31° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.01 }{ 33.37 } = 3.09 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 32 }{ 2 * sin 132° 10'18" } = 21.59 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 22.24**2 - 32**2 } }{ 2 } = 8.33 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.24**2+2 * 32**2 - 12.5**2 } }{ 2 } = 26.836 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 32**2 - 22.24**2 } }{ 2 } = 21.599 ; ;
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