Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 478.625   b = 477.937   c = 223.306596624

Area: T = 51925.75328293
Perimeter: p = 1179.868796624
Semiperimeter: s = 589.934398312

Angle ∠ A = α = 76.6721649645° = 76°40'18″ = 1.33881727292 rad
Angle ∠ B = β = 76.3288350355° = 76°19'42″ = 1.33221810263 rad
Angle ∠ C = γ = 27° = 0.4711238898 rad

Height: ha = 216.9798857474
Height: hb = 217.2911202938
Height: hc = 465.0643730304

Median: ma = 286.1376667708
Median: mb = 286.9987872116
Median: mc = 465.0666064831

Inradius: r = 88.02195993366
Circumradius: R = 245.9376827277

Vertex coordinates: A[223.306596624; 0] B[0; 0] C[113.1276554711; 465.0643730304]
Centroid: CG[112.144417365; 155.0211243435]
Coordinates of the circumscribed circle: U[111.653298312; 219.1311317642]
Coordinates of the inscribed circle: I[111.997698312; 88.02195993366]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.3288350355° = 103°19'42″ = 1.33881727292 rad
∠ B' = β' = 103.6721649645° = 103°40'18″ = 1.33221810263 rad
∠ C' = γ' = 153° = 0.4711238898 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 = 478.63 ; ; b = 477.94 ; ; gamma = 27° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 478.63**2+477.94**2 - 2 * 478.63 * 477.94 * cos(27° ) } ; ; c = 223.31 ; ;


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 = 478.63 ; ; b = 477.94 ; ; c = 223.31 ; ;

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

p = a+b+c = 478.63+477.94+223.31 = 1179.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1179.87 }{ 2 } = 589.93 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 589.93 * (589.93-478.63)(589.93-477.94)(589.93-223.31) } ; ; T = sqrt{ 2696283806.88 } = 51925.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 51925.75 }{ 478.63 } = 216.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 51925.75 }{ 477.94 } = 217.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 51925.75 }{ 223.31 } = 465.06 ; ;

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{ 478.63**2-477.94**2-223.31**2 }{ 2 * 477.94 * 223.31 } ) = 76° 40'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 477.94**2-478.63**2-223.31**2 }{ 2 * 478.63 * 223.31 } ) = 76° 19'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 223.31**2-478.63**2-477.94**2 }{ 2 * 477.94 * 478.63 } ) = 27° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 51925.75 }{ 589.93 } = 88.02 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 478.63 }{ 2 * sin 76° 40'18" } = 245.94 ; ;




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