Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 10.05498756211   b = 9.22195444573   c = 7.21111025509

Area: T = 32
Perimeter: p = 26.48105226293
Semiperimeter: s = 13.24402613147

Angle ∠ A = α = 74.2911362171° = 74°17'29″ = 1.29766288757 rad
Angle ∠ B = β = 62.02105256115° = 62°1'14″ = 1.08224623757 rad
Angle ∠ C = γ = 43.68881122175° = 43°41'17″ = 0.76325014022 rad

Height: ha = 6.36882380173
Height: hb = 6.94217746502
Height: hc = 8.87552031396

Median: ma = 6.5766473219
Median: mb = 7.43330343737
Median: mc = 8.944427191

Inradius: r = 2.41768707278
Circumradius: R = 5.22198960194

Vertex coordinates: A[2; 5] B[6; -1] C[-4; -2]
Centroid: CG[1.33333333333; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.28439625741; 2.41768707278]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.7098637829° = 105°42'31″ = 1.29766288757 rad
∠ B' = β' = 117.9799474388° = 117°58'46″ = 1.08224623757 rad
∠ C' = γ' = 136.3121887783° = 136°18'43″ = 0.76325014022 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{ (6-(-4))**2 + (-1-(-2))**2 } ; ; a = sqrt{ 101 } = 10.05 ; ;

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{ (2-(-4))**2 + (5-(-2))**2 } ; ; b = sqrt{ 85 } = 9.22 ; ;

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{ (2-6)**2 + (5-(-1))**2 } ; ; c = sqrt{ 52 } = 7.21 ; ;


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 = 10.05 ; ; b = 9.22 ; ; c = 7.21 ; ;

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

p = a+b+c = 10.05+9.22+7.21 = 26.48 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.48 }{ 2 } = 13.24 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.24 * (13.24-10.05)(13.24-9.22)(13.24-7.21) } ; ; T = sqrt{ 1024 } = 32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 32 }{ 10.05 } = 6.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 32 }{ 9.22 } = 6.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 32 }{ 7.21 } = 8.88 ; ;

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{ 10.05**2-9.22**2-7.21**2 }{ 2 * 9.22 * 7.21 } ) = 74° 17'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.22**2-10.05**2-7.21**2 }{ 2 * 10.05 * 7.21 } ) = 62° 1'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.21**2-10.05**2-9.22**2 }{ 2 * 9.22 * 10.05 } ) = 43° 41'17" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 32 }{ 13.24 } = 2.42 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.05 }{ 2 * sin 74° 17'29" } = 5.22 ; ;




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