Triangle calculator

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

You have entered side a, c and angle β.

Right scalene triangle.

Sides: a = 68.29   b = 144.6376696934   c = 127.5

Area: T = 4353.48875
Perimeter: p = 340.4276696934
Semiperimeter: s = 170.2133348467

Angle ∠ A = α = 28.17438511597° = 28°10'26″ = 0.49217264657 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 61.82661488403° = 61°49'34″ = 1.07990698611 rad

Height: ha = 127.5
Height: hb = 60.19989341887
Height: hc = 68.29

Median: ma = 131.9932920359
Median: mb = 72.3188348467
Median: mc = 93.42215531877

Inradius: r = 25.5776651533
Circumradius: R = 72.3188348467

Vertex coordinates: A[127.5; 0] B[0; 0] C[-0; 68.29]
Centroid: CG[42.5; 22.76333333333]
Coordinates of the circumscribed circle: U[63.75; 34.145]
Coordinates of the inscribed circle: I[25.5776651533; 25.5776651533]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.826614884° = 151°49'34″ = 0.49217264657 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 118.174385116° = 118°10'26″ = 1.07990698611 rad

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How did we calculate this triangle?

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

a = 68.29 ; ; c = 127.5 ; ; beta = 90° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 68.29**2+127.5**2 - 2 * 68.29 * 127.5 * cos 90° } ; ; b = 144.64 ; ;
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 = 68.29 ; ; b = 144.64 ; ; c = 127.5 ; ;

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

p = a+b+c = 68.29+144.64+127.5 = 340.43 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 340.43 }{ 2 } = 170.21 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 170.21 * (170.21-68.29)(170.21-144.64)(170.21-127.5) } ; ; T = sqrt{ 18952853.41 } = 4353.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4353.49 }{ 68.29 } = 127.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4353.49 }{ 144.64 } = 60.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4353.49 }{ 127.5 } = 68.29 ; ;

7. 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{ 144.64**2+127.5**2-68.29**2 }{ 2 * 144.64 * 127.5 } ) = 28° 10'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 68.29**2+127.5**2-144.64**2 }{ 2 * 68.29 * 127.5 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 28° 10'26" - 90° = 61° 49'34" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4353.49 }{ 170.21 } = 25.58 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 68.29 }{ 2 * sin 28° 10'26" } = 72.32 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 144.64**2+2 * 127.5**2 - 68.29**2 } }{ 2 } = 131.993 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 127.5**2+2 * 68.29**2 - 144.64**2 } }{ 2 } = 72.318 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 144.64**2+2 * 68.29**2 - 127.5**2 } }{ 2 } = 93.422 ; ;
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