Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 21.93217121995   b = 68.73986354243   c = 75.60442326857

Area: T = 742.9176549822
Perimeter: p = 166.275458031
Semiperimeter: s = 83.13772901548

Angle ∠ A = α = 16.61330020417° = 16°36'47″ = 0.29899515843 rad
Angle ∠ B = β = 63.64987152482° = 63°38'55″ = 1.11108796457 rad
Angle ∠ C = γ = 99.738828271° = 99°44'18″ = 1.74107614236 rad

Height: ha = 67.74881578334
Height: hb = 21.61656909498
Height: hc = 19.65327766616

Median: ma = 71.41660346141
Median: mb = 43.78664134179
Median: mc = 34.2643683398

Inradius: r = 8.93660207488
Circumradius: R = 38.35547830179

Vertex coordinates: A[0; 30; 0] B[60; 76; 0] C[60; 60; 15]
Centroid: CG[40; 55.33333333333; 5]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.3876997958° = 163°23'13″ = 0.29899515843 rad
∠ B' = β' = 116.3511284752° = 116°21'5″ = 1.11108796457 rad
∠ C' = γ' = 80.262171729° = 80°15'42″ = 1.74107614236 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{ (60-60)**2 + (76-60)**2 + (0 - 15)**2 } ; ; a = sqrt{ 481 } = 21.93 ; ;

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{ (0-60)**2 + (30-60)**2 + (0 - 15)**2 } ; ; b = sqrt{ 4725 } = 68.74 ; ;

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{ (0-60)**2 + (30-76)**2 + (0 - 0)**2 } ; ; c = sqrt{ 5716 } = 75.6 ; ;


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 = 21.93 ; ; b = 68.74 ; ; c = 75.6 ; ;

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

p = a+b+c = 21.93+68.74+75.6 = 166.27 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 166.27 }{ 2 } = 83.14 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 83.14 * (83.14-21.93)(83.14-68.74)(83.14-75.6) } ; ; T = sqrt{ 551925 } = 742.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 742.92 }{ 21.93 } = 67.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 742.92 }{ 68.74 } = 21.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 742.92 }{ 75.6 } = 19.65 ; ;

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{ 21.93**2-68.74**2-75.6**2 }{ 2 * 68.74 * 75.6 } ) = 16° 36'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 68.74**2-21.93**2-75.6**2 }{ 2 * 21.93 * 75.6 } ) = 63° 38'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 75.6**2-21.93**2-68.74**2 }{ 2 * 68.74 * 21.93 } ) = 99° 44'18" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 742.92 }{ 83.14 } = 8.94 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21.93 }{ 2 * sin 16° 36'47" } = 38.35 ; ;




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