Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 16.24880768093   b = 28.19657443597   c = 12.1244355653

Area: T = 22.0455407685
Perimeter: p = 56.5688176822
Semiperimeter: s = 28.2844088411

Angle ∠ A = α = 7.41103717247° = 7°24'37″ = 0.12993353854 rad
Angle ∠ B = β = 167.0676845508° = 167°4'1″ = 2.9165866525 rad
Angle ∠ C = γ = 5.52327827672° = 5°31'22″ = 0.09663907432 rad

Height: ha = 2.71436021012
Height: hb = 1.56437400739
Height: hc = 3.63765491604

Median: ma = 20.12546117975
Median: mb = 2.59880762114
Median: mc = 22.19879728804

Inradius: r = 0.77994279018
Circumradius: R = 62.98991966046

Vertex coordinates: A[1; 4; -7] B[2; -1; 4] C[0; -9; 18]
Centroid: CG[1; -2; 5]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172.5989628275° = 172°35'23″ = 0.12993353854 rad
∠ B' = β' = 12.93331544919° = 12°55'59″ = 2.9165866525 rad
∠ C' = γ' = 174.4777217233° = 174°28'38″ = 0.09663907432 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{ (2-0)**2 + (-1-(-9))**2 + (4 - 18)**2 } ; ; a = sqrt{ 264 } = 16.25 ; ;

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{ (1-0)**2 + (4-(-9))**2 + (-7 - 18)**2 } ; ; b = sqrt{ 795 } = 28.2 ; ;

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{ (1-2)**2 + (4-(-1))**2 + (-7 - 4)**2 } ; ; c = sqrt{ 147 } = 12.12 ; ;


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 = 16.25 ; ; b = 28.2 ; ; c = 12.12 ; ;

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

p = a+b+c = 16.25+28.2+12.12 = 56.57 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56.57 }{ 2 } = 28.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.28 * (28.28-16.25)(28.28-28.2)(28.28-12.12) } ; ; T = sqrt{ 486 } = 22.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.05 }{ 16.25 } = 2.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.05 }{ 28.2 } = 1.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.05 }{ 12.12 } = 3.64 ; ;

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{ 16.25**2-28.2**2-12.12**2 }{ 2 * 28.2 * 12.12 } ) = 7° 24'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.2**2-16.25**2-12.12**2 }{ 2 * 16.25 * 12.12 } ) = 167° 4'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.12**2-16.25**2-28.2**2 }{ 2 * 28.2 * 16.25 } ) = 5° 31'22" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.05 }{ 28.28 } = 0.78 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.25 }{ 2 * sin 7° 24'37" } = 62.99 ; ;




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