Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.24662112512   b = 15.62204993518   c = 12.64991106407

Area: T = 52
Perimeter: p = 36.51658212437
Semiperimeter: s = 18.25879106219

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

Height: ha = 12.61218525019
Height: hb = 6.65879177565
Height: hc = 8.22219219164

Median: ma = 13.60114705087
Median: mb = 7.28801098893
Median: mc = 10.77703296143

Inradius: r = 2.84880805431
Circumradius: R = 7.83333228419

Vertex coordinates: A[-5; -6] B[7; -2] C[5; 6]
Centroid: CG[2.33333333333; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.21990831187; 2.84880805431]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.2410519915° = 148°14'26″ = 0.55443074962 rad
∠ B' = β' = 85.6011294645° = 85°36'5″ = 1.64875682181 rad
∠ C' = γ' = 126.158818544° = 126°9'29″ = 0.94397169393 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{ (7-5)**2 + (-2-6)**2 } ; ; a = sqrt{ 68 } = 8.25 ; ;

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{ (-5-5)**2 + (-6-6)**2 } ; ; b = sqrt{ 244 } = 15.62 ; ;

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{ (-5-7)**2 + (-6-(-2))**2 } ; ; c = sqrt{ 160 } = 12.65 ; ;


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 = 8.25 ; ; b = 15.62 ; ; c = 12.65 ; ;

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

p = a+b+c = 8.25+15.62+12.65 = 36.52 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.52 }{ 2 } = 18.26 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.26 * (18.26-8.25)(18.26-15.62)(18.26-12.65) } ; ; T = sqrt{ 2704 } = 52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52 }{ 8.25 } = 12.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52 }{ 15.62 } = 6.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52 }{ 12.65 } = 8.22 ; ;

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{ 8.25**2-15.62**2-12.65**2 }{ 2 * 15.62 * 12.65 } ) = 31° 45'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.62**2-8.25**2-12.65**2 }{ 2 * 8.25 * 12.65 } ) = 94° 23'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.65**2-8.25**2-15.62**2 }{ 2 * 15.62 * 8.25 } ) = 53° 50'31" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 52 }{ 18.26 } = 2.85 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.25 }{ 2 * sin 31° 45'34" } = 7.83 ; ;




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