Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 6.32545553203   b = 11.18803398875   c = 9.84988578018

Area: T = 31
Perimeter: p = 27.35437530096
Semiperimeter: s = 13.67768765048

Angle ∠ A = α = 34.26773354433° = 34°16'2″ = 0.59880778294 rad
Angle ∠ B = β = 84.47224598483° = 84°28'21″ = 1.47443225516 rad
Angle ∠ C = γ = 61.26602047083° = 61°15'37″ = 1.06991922726 rad

Height: ha = 9.80330607465
Height: hb = 5.54554485842
Height: hc = 6.29551462238

Median: ma = 10.05498756211
Median: mb = 6.10332778079
Median: mc = 7.63221687612

Inradius: r = 2.26765993942
Circumradius: R = 5.61662855956

Vertex coordinates: A[0; -3] B[-4; 6] C[2; 8]
Centroid: CG[-0.66766666667; 3.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.21993483285; 2.26765993942]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.7332664557° = 145°43'58″ = 0.59880778294 rad
∠ B' = β' = 95.52875401517° = 95°31'39″ = 1.47443225516 rad
∠ C' = γ' = 118.7439795292° = 118°44'23″ = 1.06991922726 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{ (-4-2)**2 + (6-8)**2 } ; ; a = sqrt{ 40 } = 6.32 ; ;

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{ (0-2)**2 + (-3-8)**2 } ; ; b = sqrt{ 125 } = 11.18 ; ;

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{ (0-(-4))**2 + (-3-6)**2 } ; ; c = sqrt{ 97 } = 9.85 ; ;


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 = 6.32 ; ; b = 11.18 ; ; c = 9.85 ; ;

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

p = a+b+c = 6.32+11.18+9.85 = 27.35 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.35 }{ 2 } = 13.68 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.68 * (13.68-6.32)(13.68-11.18)(13.68-9.85) } ; ; T = sqrt{ 961 } = 31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31 }{ 6.32 } = 9.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31 }{ 11.18 } = 5.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31 }{ 9.85 } = 6.3 ; ;

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{ 6.32**2-11.18**2-9.85**2 }{ 2 * 11.18 * 9.85 } ) = 34° 16'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.18**2-6.32**2-9.85**2 }{ 2 * 6.32 * 9.85 } ) = 84° 28'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.85**2-6.32**2-11.18**2 }{ 2 * 11.18 * 6.32 } ) = 61° 15'37" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31 }{ 13.68 } = 2.27 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.32 }{ 2 * sin 34° 16'2" } = 5.62 ; ;




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