Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 14.31878210633   b = 18.02877563773   c = 13.03884048104

Area: T = 92.5
Perimeter: p = 45.3843982251
Semiperimeter: s = 22.69219911255

Angle ∠ A = α = 51.9111227119° = 51°54'40″ = 0.9066021832 rad
Angle ∠ B = β = 82.3043948278° = 82°18'14″ = 1.43664748848 rad
Angle ∠ C = γ = 45.7854824603° = 45°47'5″ = 0.79990959368 rad

Height: ha = 12.92109604717
Height: hb = 10.26219536302
Height: hc = 14.18988522937

Median: ma = 14.00989257261
Median: mb = 10.3087764064
Median: mc = 14.91664338902

Inradius: r = 4.07663280529
Circumradius: R = 9.09658093241

Vertex coordinates: A[-9; -5] B[-8; 8] C[6; 5]
Centroid: CG[-3.66766666667; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.55108551423; 4.07663280529]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.0898772881° = 128°5'20″ = 0.9066021832 rad
∠ B' = β' = 97.6966051722° = 97°41'46″ = 1.43664748848 rad
∠ C' = γ' = 134.2155175397° = 134°12'55″ = 0.79990959368 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{ (-8-6)**2 + (8-5)**2 } ; ; a = sqrt{ 205 } = 14.32 ; ;

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{ (-9-6)**2 + (-5-5)**2 } ; ; b = sqrt{ 325 } = 18.03 ; ;

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{ (-9-(-8))**2 + (-5-8)**2 } ; ; c = sqrt{ 170 } = 13.04 ; ;


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 = 14.32 ; ; b = 18.03 ; ; c = 13.04 ; ;

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

p = a+b+c = 14.32+18.03+13.04 = 45.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45.38 }{ 2 } = 22.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.69 * (22.69-14.32)(22.69-18.03)(22.69-13.04) } ; ; T = sqrt{ 8556.25 } = 92.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 92.5 }{ 14.32 } = 12.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 92.5 }{ 18.03 } = 10.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 92.5 }{ 13.04 } = 14.19 ; ;

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{ 14.32**2-18.03**2-13.04**2 }{ 2 * 18.03 * 13.04 } ) = 51° 54'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18.03**2-14.32**2-13.04**2 }{ 2 * 14.32 * 13.04 } ) = 82° 18'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.04**2-14.32**2-18.03**2 }{ 2 * 18.03 * 14.32 } ) = 45° 47'5" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 92.5 }{ 22.69 } = 4.08 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14.32 }{ 2 * sin 51° 54'40" } = 9.1 ; ;




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