Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 8.5   b = 7.7   c = 13.67697011841

Area: T = 29.65989223308
Perimeter: p = 29.87697011841
Semiperimeter: s = 14.9354850592

Angle ∠ A = α = 34.30219457746° = 34°18'7″ = 0.59986818936 rad
Angle ∠ B = β = 30.69880542254° = 30°41'53″ = 0.53657821202 rad
Angle ∠ C = γ = 115° = 2.00771286398 rad

Height: ha = 6.97985699602
Height: hb = 7.70436161898
Height: hc = 4.33993665935

Median: ma = 10.24875785057
Median: mb = 10.71113428304
Median: mc = 4.36551823999

Inradius: r = 1.98658867786
Circumradius: R = 7.54114232227

Vertex coordinates: A[13.67697011841; 0] B[0; 0] C[7.30988916784; 4.33993665935]
Centroid: CG[6.99328642875; 1.44664555312]
Coordinates of the circumscribed circle: U[6.8354850592; -3.18771431734]
Coordinates of the inscribed circle: I[7.2354850592; 1.98658867786]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.6988054225° = 145°41'53″ = 0.59986818936 rad
∠ B' = β' = 149.3021945775° = 149°18'7″ = 0.53657821202 rad
∠ C' = γ' = 65° = 2.00771286398 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 = 8.5 ; ; b = 7.7 ; ; gamma = 115° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 8.5**2+7.7**2 - 2 * 8.5 * 7.7 * cos(115° ) } ; ; c = 13.67 ; ;


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 = 8.5 ; ; b = 7.7 ; ; c = 13.67 ; ;

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

p = a+b+c = 8.5+7.7+13.67 = 29.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.87 }{ 2 } = 14.93 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.93 * (14.93-8.5)(14.93-7.7)(14.93-13.67) } ; ; T = sqrt{ 879.65 } = 29.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.66 }{ 8.5 } = 6.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.66 }{ 7.7 } = 7.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.66 }{ 13.67 } = 4.34 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8.5**2-7.7**2-13.67**2 }{ 2 * 7.7 * 13.67 } ) = 34° 18'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.7**2-8.5**2-13.67**2 }{ 2 * 8.5 * 13.67 } ) = 30° 41'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.67**2-8.5**2-7.7**2 }{ 2 * 7.7 * 8.5 } ) = 115° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.66 }{ 14.93 } = 1.99 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.5 }{ 2 * sin 34° 18'7" } = 7.54 ; ;




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