Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 4.12331056256   b = 6.32545553203   c = 7.81102496759

Area: T = 13
Perimeter: p = 18.25879106219
Semiperimeter: s = 9.12989553109

Angle ∠ A = α = 31.75994800848° = 31°45'34″ = 0.55443074962 rad
Angle ∠ B = β = 53.84218145602° = 53°50'31″ = 0.94397169393 rad
Angle ∠ C = γ = 94.3998705355° = 94°23'55″ = 1.64875682181 rad

Height: ha = 6.30659262509
Height: hb = 4.11109609582
Height: hc = 3.32989588783

Median: ma = 6.80107352544
Median: mb = 5.38551648071
Median: mc = 3.64400549446

Inradius: r = 1.42440402716
Circumradius: R = 3.9176661421

Vertex coordinates: A[-4; 3] B[1; -3] C[2; 1]
Centroid: CG[-0.33333333333; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.04106448138; 1.42440402716]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.2410519915° = 148°14'26″ = 0.55443074962 rad
∠ B' = β' = 126.158818544° = 126°9'29″ = 0.94397169393 rad
∠ C' = γ' = 85.6011294645° = 85°36'5″ = 1.64875682181 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{ (1-2)**2 + (-3-1)**2 } ; ; a = sqrt{ 17 } = 4.12 ; ;

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{ (-4-2)**2 + (3-1)**2 } ; ; b = sqrt{ 40 } = 6.32 ; ;

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{ (-4-1)**2 + (3-(-3))**2 } ; ; c = sqrt{ 61 } = 7.81 ; ;


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 = 4.12 ; ; b = 6.32 ; ; c = 7.81 ; ;

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

p = a+b+c = 4.12+6.32+7.81 = 18.26 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.26 }{ 2 } = 9.13 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.13 * (9.13-4.12)(9.13-6.32)(9.13-7.81) } ; ; T = sqrt{ 169 } = 13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13 }{ 4.12 } = 6.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13 }{ 6.32 } = 4.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13 }{ 7.81 } = 3.33 ; ;

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{ 4.12**2-6.32**2-7.81**2 }{ 2 * 6.32 * 7.81 } ) = 31° 45'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.32**2-4.12**2-7.81**2 }{ 2 * 4.12 * 7.81 } ) = 53° 50'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.81**2-4.12**2-6.32**2 }{ 2 * 6.32 * 4.12 } ) = 94° 23'55" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13 }{ 9.13 } = 1.42 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.12 }{ 2 * sin 31° 45'34" } = 3.92 ; ;




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