Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 87.00657469366   b = 92.91439386745   c = 68.24222156733

Area: T = 2829.5
Perimeter: p = 248.1621901284
Semiperimeter: s = 124.0810950642

Angle ∠ A = α = 63.18884386674° = 63°11'18″ = 1.10328463039 rad
Angle ∠ B = β = 72.38328963895° = 72°22'58″ = 1.26333198641 rad
Angle ∠ C = γ = 44.42986649431° = 44°25'43″ = 0.77554264855 rad

Height: ha = 65.04216805699
Height: hb = 60.90658240425
Height: hc = 82.92552090391

Median: ma = 68.93883782809
Median: mb = 62.89107783383
Median: mc = 83.29901554807

Inradius: r = 22.80436615238
Circumradius: R = 48.74329982322

Vertex coordinates: A[707; 518] B[668; 574] C[615; 505]
Centroid: CG[663.3333333333; 532.3333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[7.24112404592; 22.80436615238]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.8121561333° = 116°48'42″ = 1.10328463039 rad
∠ B' = β' = 107.617710361° = 107°37'2″ = 1.26333198641 rad
∠ C' = γ' = 135.5711335057° = 135°34'17″ = 0.77554264855 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{ (668-615)**2 + (574-505)**2 } ; ; a = sqrt{ 7570 } = 87.01 ; ;

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{ (707-615)**2 + (518-505)**2 } ; ; b = sqrt{ 8633 } = 92.91 ; ;

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{ (707-668)**2 + (518-574)**2 } ; ; c = sqrt{ 4657 } = 68.24 ; ;


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 = 87.01 ; ; b = 92.91 ; ; c = 68.24 ; ;

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

p = a+b+c = 87.01+92.91+68.24 = 248.16 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 248.16 }{ 2 } = 124.08 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 124.08 * (124.08-87.01)(124.08-92.91)(124.08-68.24) } ; ; T = sqrt{ 8006070.25 } = 2829.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2829.5 }{ 87.01 } = 65.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2829.5 }{ 92.91 } = 60.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2829.5 }{ 68.24 } = 82.93 ; ;

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{ 87.01**2-92.91**2-68.24**2 }{ 2 * 92.91 * 68.24 } ) = 63° 11'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 92.91**2-87.01**2-68.24**2 }{ 2 * 87.01 * 68.24 } ) = 72° 22'58" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 68.24**2-87.01**2-92.91**2 }{ 2 * 92.91 * 87.01 } ) = 44° 25'43" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2829.5 }{ 124.08 } = 22.8 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 87.01 }{ 2 * sin 63° 11'18" } = 48.74 ; ;




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