Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 31.95330906173   b = 35.60989876295   c = 11.18803398875

Area: T = 176
Perimeter: p = 78.74224181344
Semiperimeter: s = 39.37112090672

Angle ∠ A = α = 62.14876198814° = 62°8'51″ = 1.08546805892 rad
Angle ∠ B = β = 99.83114569595° = 99°49'53″ = 1.7422387621 rad
Angle ∠ C = γ = 18.02109231591° = 18°1'15″ = 0.31545244434 rad

Height: ha = 11.01661487731
Height: hb = 9.88551448309
Height: hc = 31.48438371232

Median: ma = 21.00659515376
Median: mb = 16
Median: mc = 33.36554012414

Inradius: r = 4.47702716571
Circumradius: R = 18.07698623879

Vertex coordinates: A[11; 4] B[13; -7] C[-17; -18]
Centroid: CG[2.33333333333; -7]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.77546777588; 4.47702716571]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.8522380119° = 117°51'9″ = 1.08546805892 rad
∠ B' = β' = 80.16985430405° = 80°10'7″ = 1.7422387621 rad
∠ C' = γ' = 161.9799076841° = 161°58'45″ = 0.31545244434 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{ (13-(-17))**2 + (-7-(-18))**2 } ; ; a = sqrt{ 1021 } = 31.95 ; ;

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-(-17))**2 + (4-(-18))**2 } ; ; b = sqrt{ 1268 } = 35.61 ; ;

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{ (11-13)**2 + (4-(-7))**2 } ; ; c = sqrt{ 125 } = 11.18 ; ;


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 = 31.95 ; ; b = 35.61 ; ; c = 11.18 ; ;

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

p = a+b+c = 31.95+35.61+11.18 = 78.74 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78.74 }{ 2 } = 39.37 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.37 * (39.37-31.95)(39.37-35.61)(39.37-11.18) } ; ; T = sqrt{ 30976 } = 176 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 176 }{ 31.95 } = 11.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 176 }{ 35.61 } = 9.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 176 }{ 11.18 } = 31.48 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 35.61**2+11.18**2-31.95**2 }{ 2 * 35.61 * 11.18 } ) = 62° 8'51" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31.95**2+11.18**2-35.61**2 }{ 2 * 31.95 * 11.18 } ) = 99° 49'53" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 31.95**2+35.61**2-11.18**2 }{ 2 * 31.95 * 35.61 } ) = 18° 1'15" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 176 }{ 39.37 } = 4.47 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 31.95 }{ 2 * sin 62° 8'51" } = 18.07 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.61**2+2 * 11.18**2 - 31.95**2 } }{ 2 } = 21.006 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.18**2+2 * 31.95**2 - 35.61**2 } }{ 2 } = 16 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.61**2+2 * 31.95**2 - 11.18**2 } }{ 2 } = 33.365 ; ;
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