Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 4.12331056256   b = 2.23660679775   c = 5.83109518948

Area: T = 3.5
Perimeter: p = 12.1990125498
Semiperimeter: s = 6.0955062749

Angle ∠ A = α = 32.47111922908° = 32°28'16″ = 0.56767292175 rad
Angle ∠ B = β = 16.92875130641° = 16°55'39″ = 0.29554408371 rad
Angle ∠ C = γ = 130.6011294645° = 130°36'5″ = 2.27994225989 rad

Height: ha = 1.69877493753
Height: hb = 3.13304951685
Height: hc = 1.2200490096

Median: ma = 3.9055124838
Median: mb = 4.92444289009
Median: mc = 1.58111388301

Inradius: r = 0.57442352695
Circumradius: R = 3.84399085873

Vertex coordinates: A[-5; 10] B[0; 7] C[-4; 8]
Centroid: CG[-3; 8.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.88767730282; 0.57442352695]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.5298807709° = 147°31'44″ = 0.56767292175 rad
∠ B' = β' = 163.0722486936° = 163°4'21″ = 0.29554408371 rad
∠ C' = γ' = 49.3998705355° = 49°23'55″ = 2.27994225989 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{ (0-(-4))**2 + (7-8)**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{ (-5-(-4))**2 + (10-8)**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (-5-0)**2 + (10-7)**2 } ; ; c = sqrt{ 34 } = 5.83 ; ;


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 = 2.24 ; ; c = 5.83 ; ;

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

p = a+b+c = 4.12+2.24+5.83 = 12.19 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.19 }{ 2 } = 6.1 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.1 * (6.1-4.12)(6.1-2.24)(6.1-5.83) } ; ; T = sqrt{ 12.25 } = 3.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.5 }{ 4.12 } = 1.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.5 }{ 2.24 } = 3.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.5 }{ 5.83 } = 1.2 ; ;

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**2-2.24**2-5.83**2 }{ 2 * 2.24 * 5.83 } ) = 32° 28'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.24**2-4.12**2-5.83**2 }{ 2 * 4.12 * 5.83 } ) = 16° 55'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.83**2-4.12**2-2.24**2 }{ 2 * 2.24 * 4.12 } ) = 130° 36'5" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.5 }{ 6.1 } = 0.57 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.12 }{ 2 * sin 32° 28'16" } = 3.84 ; ;




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