Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 519.1398709788   b = 325.0110769052   c = 200.9900472871

Area: T = 10602
Perimeter: p = 1045.054995171
Semiperimeter: s = 522.5254975856

Angle ∠ A = α = 161.0550026851° = 161°3' = 2.8110853229 rad
Angle ∠ B = β = 11.73304402549° = 11°43'50″ = 0.20547348052 rad
Angle ∠ C = γ = 7.22195328941° = 7°13'10″ = 0.12660046195 rad

Height: ha = 40.84545750629
Height: hb = 65.24109151298
Height: hc = 105.545479886

Median: ma = 74.96883266453
Median: mb = 358.5043835405
Median: mc = 421.2821675367

Inradius: r = 20.2989939218
Circumradius: R = 799.3088318061

Vertex coordinates: A[76; 316] B[57; 516] C[0; 0]
Centroid: CG[44.33333333333; 277.3333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[97.71549463839; 20.2989939218]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 18.9549973149° = 18°57' = 2.8110853229 rad
∠ B' = β' = 168.2769559745° = 168°16'10″ = 0.20547348052 rad
∠ C' = γ' = 172.7880467106° = 172°46'50″ = 0.12660046195 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{ (57-0)**2 + (516-0)**2 } ; ; a = sqrt{ 269505 } = 519.14 ; ;

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{ (76-0)**2 + (316-0)**2 } ; ; b = sqrt{ 105632 } = 325.01 ; ;

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{ (76-57)**2 + (316-516)**2 } ; ; c = sqrt{ 40361 } = 200.9 ; ;


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 = 519.14 ; ; b = 325.01 ; ; c = 200.9 ; ;

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

p = a+b+c = 519.14+325.01+200.9 = 1045.05 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1045.05 }{ 2 } = 522.52 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 522.52 * (522.52-519.14)(522.52-325.01)(522.52-200.9) } ; ; T = sqrt{ 112402404 } = 10602 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10602 }{ 519.14 } = 40.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10602 }{ 325.01 } = 65.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10602 }{ 200.9 } = 105.54 ; ;

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{ 519.14**2-325.01**2-200.9**2 }{ 2 * 325.01 * 200.9 } ) = 161° 3' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 325.01**2-519.14**2-200.9**2 }{ 2 * 519.14 * 200.9 } ) = 11° 43'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200.9**2-519.14**2-325.01**2 }{ 2 * 325.01 * 519.14 } ) = 7° 13'10" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10602 }{ 522.52 } = 20.29 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 519.14 }{ 2 * sin 161° 3' } = 799.31 ; ;




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