Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 16.49224225025   b = 24.16660919472   c = 15.23215462117

Area: T = 124
Perimeter: p = 55.89900606614
Semiperimeter: s = 27.94550303307

Angle ∠ A = α = 42.35774547059° = 42°21'27″ = 0.73992770474 rad
Angle ∠ B = β = 99.16223470457° = 99°9'44″ = 1.731070945 rad
Angle ∠ C = γ = 38.48801982483° = 38°28'49″ = 0.67216061563 rad

Height: ha = 15.03772087523
Height: hb = 10.26223130187
Height: hc = 16.28219976746

Median: ma = 18.43990889146
Median: mb = 10.2965630141
Median: mc = 19.23553840617

Inradius: r = 4.4377282713
Circumradius: R = 12.23992045065

Vertex coordinates: A[-11; -2; 0] B[-5; 12; 0] C[11; 8; 0]
Centroid: CG[-1.66766666667; 6; 0]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.6432545294° = 137°38'33″ = 0.73992770474 rad
∠ B' = β' = 80.83876529543° = 80°50'16″ = 1.731070945 rad
∠ C' = γ' = 141.5219801752° = 141°31'11″ = 0.67216061563 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{ (-5-11)**2 + (12-8)**2 } ; ; a = sqrt{ 272 } = 16.49 ; ;

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{ (-11-11)**2 + (-2-8)**2 } ; ; b = sqrt{ 584 } = 24.17 ; ;

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{ (-11-(-5))**2 + (-2-12)**2 } ; ; c = sqrt{ 232 } = 15.23 ; ;


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 = 16.49 ; ; b = 24.17 ; ; c = 15.23 ; ;

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

p = a+b+c = 16.49+24.17+15.23 = 55.89 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55.89 }{ 2 } = 27.95 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.95 * (27.95-16.49)(27.95-24.17)(27.95-15.23) } ; ; T = sqrt{ 15376 } = 124 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 124 }{ 16.49 } = 15.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 124 }{ 24.17 } = 10.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 124 }{ 15.23 } = 16.28 ; ;

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{ 16.49**2-24.17**2-15.23**2 }{ 2 * 24.17 * 15.23 } ) = 42° 21'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24.17**2-16.49**2-15.23**2 }{ 2 * 16.49 * 15.23 } ) = 99° 9'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.23**2-16.49**2-24.17**2 }{ 2 * 24.17 * 16.49 } ) = 38° 28'49" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 124 }{ 27.95 } = 4.44 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.49 }{ 2 * sin 42° 21'27" } = 12.24 ; ;




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