Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 9.22195444573   b = 11.4021754251   c = 6.40331242374

Area: T = 29.5
Perimeter: p = 27.02444229457
Semiperimeter: s = 13.51222114729

Angle ∠ A = α = 53.91549269571° = 53°54'54″ = 0.94109929914 rad
Angle ∠ B = β = 91.94114863909° = 91°56'29″ = 1.60546816567 rad
Angle ∠ C = γ = 34.14435866519° = 34°8'37″ = 0.59659180055 rad

Height: ha = 6.39994485057
Height: hb = 5.17546423139
Height: hc = 9.21442519514

Median: ma = 8.01656097709
Median: mb = 5.52326805086
Median: mc = 9.86215414617

Inradius: r = 2.18332103545
Circumradius: R = 5.704415161

Vertex coordinates: A[-3; 2] B[2; 6] C[8; -1]
Centroid: CG[2.33333333333; 2.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.07440071307; 2.18332103545]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.0855073043° = 126°5'6″ = 0.94109929914 rad
∠ B' = β' = 88.05985136091° = 88°3'31″ = 1.60546816567 rad
∠ C' = γ' = 145.8566413348° = 145°51'23″ = 0.59659180055 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{ (2-8)**2 + (6-(-1))**2 } ; ; a = sqrt{ 85 } = 9.22 ; ;

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{ (-3-8)**2 + (2-(-1))**2 } ; ; b = sqrt{ 130 } = 11.4 ; ;

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{ (-3-2)**2 + (2-6)**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 9.22 ; ; b = 11.4 ; ; c = 6.4 ; ;

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

p = a+b+c = 9.22+11.4+6.4 = 27.02 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.02 }{ 2 } = 13.51 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.51 * (13.51-9.22)(13.51-11.4)(13.51-6.4) } ; ; T = sqrt{ 870.25 } = 29.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.5 }{ 9.22 } = 6.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.5 }{ 11.4 } = 5.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.5 }{ 6.4 } = 9.21 ; ;

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{ 9.22**2-11.4**2-6.4**2 }{ 2 * 11.4 * 6.4 } ) = 53° 54'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.4**2-9.22**2-6.4**2 }{ 2 * 9.22 * 6.4 } ) = 91° 56'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.4**2-9.22**2-11.4**2 }{ 2 * 11.4 * 9.22 } ) = 34° 8'37" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.5 }{ 13.51 } = 2.18 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.22 }{ 2 * sin 53° 54'54" } = 5.7 ; ;




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