Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side c, angle β and angle γ.

Obtuse scalene triangle.

Sides: a = 187.0343518004   b = 21.28795032022   c = 200

Area: T = 1630.105451804
Perimeter: p = 408.3133021207
Semiperimeter: s = 204.1576510603

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 5° = 0.08772664626 rad
Angle ∠ C = γ = 125° = 2.1821661565 rad

Height: ha = 17.43111485495
Height: hb = 153.2098888624
Height: hc = 16.30110451804

Median: ma = 107.1549542295
Median: mb = 193.3332780755
Median: mc = 87.84774647147

Inradius: r = 7.98545825794
Circumradius: R = 122.0777458876

Vertex coordinates: A[200; 0] B[0; 0] C[186.3221799001; 16.30110451804]
Centroid: CG[128.7743933; 5.43436817268]
Coordinates of the circumscribed circle: U[100; -70.0210753821]
Coordinates of the inscribed circle: I[182.8777007401; 7.98545825794]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 175° = 0.08772664626 rad
∠ C' = γ' = 55° = 2.1821661565 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle β and angle γ.

c = 200 ; ; beta = 5° ; ; gamma = 125° ; ;

2. From angle β, angle γ and side c we calculate b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 200 * fraction{ sin(5° ) }{ sin (125° ) } = 21.28 ; ;

3. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 21.28**2+200**2 - 2 * 21.28 * 200 * cos(50° ) } ; ; a = 187.03 ; ;


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 = 187.03 ; ; b = 21.28 ; ; c = 200 ; ;

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

p = a+b+c = 187.03+21.28+200 = 408.31 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 408.31 }{ 2 } = 204.16 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 204.16 * (204.16-187.03)(204.16-21.28)(204.16-200) } ; ; T = sqrt{ 2657240.74 } = 1630.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1630.1 }{ 187.03 } = 17.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1630.1 }{ 21.28 } = 153.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1630.1 }{ 200 } = 16.3 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 187.03**2-21.28**2-200**2 }{ 2 * 21.28 * 200 } ) = 50° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.28**2-187.03**2-200**2 }{ 2 * 187.03 * 200 } ) = 5° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200**2-187.03**2-21.28**2 }{ 2 * 21.28 * 187.03 } ) = 125° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1630.1 }{ 204.16 } = 7.98 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 187.03 }{ 2 * sin 50° } = 122.08 ; ;




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