Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 11.18803398875   b = 12.53299640861   c = 21.4010934559

Area: T = 54.5
Perimeter: p = 45.11112385327
Semiperimeter: s = 22.55656192663

Angle ∠ A = α = 23.98441837026° = 23°59'3″ = 0.4198602974 rad
Angle ∠ B = β = 27.10105101626° = 27°6'2″ = 0.47329931313 rad
Angle ∠ C = γ = 128.9155306135° = 128°54'55″ = 2.25499965483 rad

Height: ha = 9.74992563819
Height: hb = 8.69991470407
Height: hc = 5.09332355173

Median: ma = 16.62107701386
Median: mb = 15.88223801743
Median: mc = 5.14878150705

Inradius: r = 2.41662493327
Circumradius: R = 13.75224817756

Vertex coordinates: A[4; 7] B[-9; -10] C[-7; 1]
Centroid: CG[-4; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[4.722166154; 2.41662493327]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.0165816297° = 156°57″ = 0.4198602974 rad
∠ B' = β' = 152.8999489837° = 152°53'58″ = 0.47329931313 rad
∠ C' = γ' = 51.08546938653° = 51°5'5″ = 2.25499965483 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{ (-9-(-7))**2 + (-10-1)**2 } ; ; a = sqrt{ 125 } = 11.18 ; ;

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-(-7))**2 + (7-1)**2 } ; ; b = sqrt{ 157 } = 12.53 ; ;

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-(-9))**2 + (7-(-10))**2 } ; ; c = sqrt{ 458 } = 21.4 ; ;


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 = 11.18 ; ; b = 12.53 ; ; c = 21.4 ; ;

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

p = a+b+c = 11.18+12.53+21.4 = 45.11 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45.11 }{ 2 } = 22.56 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.56 * (22.56-11.18)(22.56-12.53)(22.56-21.4) } ; ; T = sqrt{ 2970.25 } = 54.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 54.5 }{ 11.18 } = 9.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 54.5 }{ 12.53 } = 8.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 54.5 }{ 21.4 } = 5.09 ; ;

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{ 11.18**2-12.53**2-21.4**2 }{ 2 * 12.53 * 21.4 } ) = 23° 59'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.53**2-11.18**2-21.4**2 }{ 2 * 11.18 * 21.4 } ) = 27° 6'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21.4**2-11.18**2-12.53**2 }{ 2 * 12.53 * 11.18 } ) = 128° 54'55" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 54.5 }{ 22.56 } = 2.42 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.18 }{ 2 * sin 23° 59'3" } = 13.75 ; ;




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