Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 21.2088267256   b = 36.51107285602   c = 33.72880817717

Area: T = 351.965953403
Perimeter: p = 91.44770775879
Semiperimeter: s = 45.7243538794

Angle ∠ A = α = 34.86435623162° = 34°51'49″ = 0.60884839514 rad
Angle ∠ B = β = 79.75993720006° = 79°45'34″ = 1.39220636507 rad
Angle ∠ C = γ = 65.37770656832° = 65°22'37″ = 1.14110450515 rad

Height: ha = 33.19107863837
Height: hb = 19.28797869508
Height: hc = 20.87704151284

Median: ma = 33.50991144317
Median: mb = 21.45876029649
Median: mc = 24.63876962194

Inradius: r = 7.69875567358
Circumradius: R = 18.55108837331

Vertex coordinates: A[-35.47; 12.78; 55.65] B[-10.68; -2.35; 38.5] C[-7.81; 17.74; 32.34]
Centroid: CG[-17.98766666667; 9.39; 42.16333333333]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.1366437684° = 145°8'11″ = 0.60884839514 rad
∠ B' = β' = 100.2410627999° = 100°14'26″ = 1.39220636507 rad
∠ C' = γ' = 114.6232934317° = 114°37'23″ = 1.14110450515 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 + ( beta _z- gamma _z)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 + ( beta _z- gamma _z)**2 } ; ; a = sqrt{ (-10.68-(-7.81))**2 + (-2.35-17.74)**2 + (38.5 - 32.34)**2 } ; ; a = sqrt{ 449.791 } = 21.21 ; ;

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 + ( alpha _z- gamma _z)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 + ( alpha _z- gamma _z)**2 } ; ; b = sqrt{ (-35.47-(-7.81))**2 + (12.78-17.74)**2 + (55.65 - 32.34)**2 } ; ; b = sqrt{ 1333.033 } = 36.51 ; ;

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 + ( alpha _z- beta _z)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 + ( alpha _z- beta _z)**2 } ; ; c = sqrt{ (-35.47-(-10.68))**2 + (12.78-(-2.35))**2 + (55.65 - 38.5)**2 } ; ; c = sqrt{ 1137.584 } = 33.73 ; ;


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 = 21.21 ; ; b = 36.51 ; ; c = 33.73 ; ;

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

p = a+b+c = 21.21+36.51+33.73 = 91.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 91.45 }{ 2 } = 45.72 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.72 * (45.72-21.21)(45.72-36.51)(45.72-33.73) } ; ; T = sqrt{ 123875.51 } = 351.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 351.96 }{ 21.21 } = 33.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 351.96 }{ 36.51 } = 19.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 351.96 }{ 33.73 } = 20.87 ; ;

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{ 21.21**2-36.51**2-33.73**2 }{ 2 * 36.51 * 33.73 } ) = 34° 51'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 36.51**2-21.21**2-33.73**2 }{ 2 * 21.21 * 33.73 } ) = 79° 45'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 33.73**2-21.21**2-36.51**2 }{ 2 * 36.51 * 21.21 } ) = 65° 22'37" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 351.96 }{ 45.72 } = 7.7 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21.21 }{ 2 * sin 34° 51'49" } = 18.55 ; ;




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