Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.38551648071   b = 7.21111025509   c = 4.12331056256

Area: T = 11
Perimeter: p = 16.71993729837
Semiperimeter: s = 8.36596864918

Angle ∠ A = α = 47.72663109939° = 47°43'35″ = 0.83329812667 rad
Angle ∠ B = β = 97.76551660184° = 97°45'55″ = 1.70663240408 rad
Angle ∠ C = γ = 34.50985229877° = 34°30'31″ = 0.60222873461 rad

Height: ha = 4.08552974399
Height: hb = 3.05108510792
Height: hc = 5.33657837508

Median: ma = 5.22201532545
Median: mb = 3.16222776602
Median: mc = 6.02107972894

Inradius: r = 1.31658388189
Circumradius: R = 3.63989195563

Vertex coordinates: A[6; 10] B[7; 6] C[2; 4]
Centroid: CG[5; 6.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.17994325662; 1.31658388189]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.2743689006° = 132°16'25″ = 0.83329812667 rad
∠ B' = β' = 82.23548339816° = 82°14'5″ = 1.70663240408 rad
∠ C' = γ' = 145.4911477012° = 145°29'29″ = 0.60222873461 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{ (7-2)**2 + (6-4)**2 } ; ; a = sqrt{ 29 } = 5.39 ; ;

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{ (6-2)**2 + (10-4)**2 } ; ; b = sqrt{ 52 } = 7.21 ; ;

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{ (6-7)**2 + (10-6)**2 } ; ; c = sqrt{ 17 } = 4.12 ; ;


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 = 5.39 ; ; b = 7.21 ; ; c = 4.12 ; ;

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

p = a+b+c = 5.39+7.21+4.12 = 16.72 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.72 }{ 2 } = 8.36 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.36 * (8.36-5.39)(8.36-7.21)(8.36-4.12) } ; ; T = sqrt{ 121 } = 11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11 }{ 5.39 } = 4.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11 }{ 7.21 } = 3.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11 }{ 4.12 } = 5.34 ; ;

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{ 5.39**2-7.21**2-4.12**2 }{ 2 * 7.21 * 4.12 } ) = 47° 43'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.21**2-5.39**2-4.12**2 }{ 2 * 5.39 * 4.12 } ) = 97° 45'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.12**2-5.39**2-7.21**2 }{ 2 * 7.21 * 5.39 } ) = 34° 30'31" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11 }{ 8.36 } = 1.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.39 }{ 2 * sin 47° 43'35" } = 3.64 ; ;




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