Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 218.5   b = 224.5   c = 215.5110350677

Area: T = 20844.99993711
Perimeter: p = 658.5110350677
Semiperimeter: s = 329.2555175339

Angle ∠ A = α = 59.50660511828° = 59°30'22″ = 1.0398576518 rad
Angle ∠ B = β = 62.29439488172° = 62°17'38″ = 1.08772345109 rad
Angle ∠ C = γ = 58.2° = 58°12' = 1.01657816247 rad

Height: ha = 190.8010909576
Height: hb = 185.7021553418
Height: hc = 193.4487779242

Median: ma = 191.0155491844
Median: mb = 185.7244037552
Median: mc = 193.5466046686

Inradius: r = 63.31095572444
Circumradius: R = 126.7876800531

Vertex coordinates: A[215.5110350677; 0] B[0; 0] C[101.588841817; 193.4487779242]
Centroid: CG[105.7699589616; 64.48325930807]
Coordinates of the circumscribed circle: U[107.7555175339; 66.81110393324]
Coordinates of the inscribed circle: I[104.7555175339; 63.31095572444]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.4943948817° = 120°29'38″ = 1.0398576518 rad
∠ B' = β' = 117.7066051183° = 117°42'22″ = 1.08772345109 rad
∠ C' = γ' = 121.8° = 121°48' = 1.01657816247 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 = 218.5 ; ; b = 224.5 ; ; gamma = 58° 12' ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 218.5**2+224.5**2 - 2 * 218.5 * 224.5 * cos(58° 12') } ; ; c = 215.51 ; ;


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 = 218.5 ; ; b = 224.5 ; ; c = 215.51 ; ;

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

p = a+b+c = 218.5+224.5+215.51 = 658.51 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 658.51 }{ 2 } = 329.26 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 329.26 * (329.26-218.5)(329.26-224.5)(329.26-215.51) } ; ; T = sqrt{ 434513998.78 } = 20845 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20845 }{ 218.5 } = 190.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20845 }{ 224.5 } = 185.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20845 }{ 215.51 } = 193.45 ; ;

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{ 218.5**2-224.5**2-215.51**2 }{ 2 * 224.5 * 215.51 } ) = 59° 30'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 224.5**2-218.5**2-215.51**2 }{ 2 * 218.5 * 215.51 } ) = 62° 17'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 215.51**2-218.5**2-224.5**2 }{ 2 * 224.5 * 218.5 } ) = 58° 12' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20845 }{ 329.26 } = 63.31 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 218.5 }{ 2 * sin 59° 30'22" } = 126.79 ; ;




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