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 = 4000   b = 6403.124423743   c = 5000

Area: T = 10000000
Perimeter: p = 15403.12442374
Semiperimeter: s = 7701.562211872

Angle ∠ A = α = 38.66598082541° = 38°39'35″ = 0.67547409422 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 51.34401917459° = 51°20'25″ = 0.89660553846 rad

Height: ha = 5000
Height: hb = 3123.475523777
Height: hc = 4000

Median: ma = 5385.165480713
Median: mb = 3201.562211872
Median: mc = 4716.991056603

Inradius: r = 1298.438788128
Circumradius: R = 3201.562211872

Vertex coordinates: A[5000; 0] B[0; 0] C[-0; 4000]
Centroid: CG[1666.667666667; 1333.333333333]
Coordinates of the circumscribed circle: U[2500; 2000]
Coordinates of the inscribed circle: I[1298.438788128; 1298.438788128]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.3440191746° = 141°20'25″ = 0.67547409422 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 128.6659808254° = 128°39'35″ = 0.89660553846 rad

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How did we calculate this triangle?

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

a = 4000 ; ; c = 5000 ; ; 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{ 4000**2+5000**2 - 2 * 4000 * 5000 * cos(90° ) } ; ; b = 6403.12 ; ;


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 = 4000 ; ; b = 6403.12 ; ; c = 5000 ; ;

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

p = a+b+c = 4000+6403.12+5000 = 15403.12 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15403.12 }{ 2 } = 7701.56 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7701.56 * (7701.56-4000)(7701.56-6403.12)(7701.56-5000) } ; ; T = sqrt{ 1 * 10**{ 14 } } = 10000000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10000000 }{ 4000 } = 5000 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10000000 }{ 6403.12 } = 3123.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10000000 }{ 5000 } = 4000 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 4000**2-6403.12**2-5000**2 }{ 2 * 6403.12 * 5000 } ) = 38° 39'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6403.12**2-4000**2-5000**2 }{ 2 * 4000 * 5000 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5000**2-4000**2-6403.12**2 }{ 2 * 6403.12 * 4000 } ) = 51° 20'25" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10000000 }{ 7701.56 } = 1298.44 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4000 }{ 2 * sin 38° 39'35" } = 3201.56 ; ;




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