Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 38.05325951809   b = 5.65768542495   c = 42.04875920833

Area: T = 80
Perimeter: p = 85.75770415136
Semiperimeter: s = 42.87985207568

Angle ∠ A = α = 42.27436890061° = 42°16'25″ = 0.73878150601 rad
Angle ∠ B = β = 5.73990984981° = 5°44'21″ = 0.11001661649 rad
Angle ∠ C = γ = 131.9877212496° = 131°59'14″ = 2.30436114286 rad

Height: ha = 4.20547066498
Height: hb = 28.28442712475
Height: hc = 3.80552119532

Median: ma = 23.19548270095
Median: mb = 40
Median: mc = 17.26326765016

Inradius: r = 1.86657360046
Circumradius: R = 28.28546247986

Vertex coordinates: A[1; 2] B[3; 44] C[5; 6]
Centroid: CG[3; 17.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[18.56440732458; 1.86657360046]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.7266310994° = 137°43'35″ = 0.73878150601 rad
∠ B' = β' = 174.2610901502° = 174°15'39″ = 0.11001661649 rad
∠ C' = γ' = 48.01327875042° = 48°46″ = 2.30436114286 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (3-5)**2 + (44-6)**2 } ; ; a = sqrt{ 1448 } = 38.05 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (1-5)**2 + (2-6)**2 } ; ; b = sqrt{ 32 } = 5.66 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (1-3)**2 + (2-44)**2 } ; ; c = sqrt{ 1768 } = 42.05 ; ;


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 = 38.05 ; ; b = 5.66 ; ; c = 42.05 ; ;

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

p = a+b+c = 38.05+5.66+42.05 = 85.76 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 85.76 }{ 2 } = 42.88 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.88 * (42.88-38.05)(42.88-5.66)(42.88-42.05) } ; ; T = sqrt{ 6400 } = 80 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80 }{ 38.05 } = 4.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80 }{ 5.66 } = 28.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80 }{ 42.05 } = 3.81 ; ;

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{ 38.05**2-5.66**2-42.05**2 }{ 2 * 5.66 * 42.05 } ) = 42° 16'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.66**2-38.05**2-42.05**2 }{ 2 * 38.05 * 42.05 } ) = 5° 44'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42.05**2-38.05**2-5.66**2 }{ 2 * 5.66 * 38.05 } ) = 131° 59'14" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80 }{ 42.88 } = 1.87 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38.05 }{ 2 * sin 42° 16'25" } = 28.28 ; ;




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