Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 17.26326765016   b = 18.43990889146   c = 9.48768329805

Area: T = 81
Perimeter: p = 45.18985983967
Semiperimeter: s = 22.59442991984

Angle ∠ A = α = 67.83436541779° = 67°50'1″ = 1.18439206091 rad
Angle ∠ B = β = 81.57330309785° = 81°34'23″ = 1.42437179714 rad
Angle ∠ C = γ = 30.59333148436° = 30°35'36″ = 0.53439540731 rad

Height: ha = 9.38444080311
Height: hb = 8.78656835417
Height: hc = 17.07662993649

Median: ma = 11.85332695911
Median: mb = 10.44403065089
Median: mc = 17.21991753577

Inradius: r = 3.58549750988
Circumradius: R = 9.32201700238

Vertex coordinates: A[4; -5] B[-5; -8] C[-8; 9]
Centroid: CG[-3; -1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.5311107422; 3.58549750988]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.1666345822° = 112°9'59″ = 1.18439206091 rad
∠ B' = β' = 98.42769690215° = 98°25'37″ = 1.42437179714 rad
∠ C' = γ' = 149.4076685156° = 149°24'24″ = 0.53439540731 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{ (-5-(-8))**2 + (-8-9)**2 } ; ; a = sqrt{ 298 } = 17.26 ; ;

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{ (4-(-8))**2 + (-5-9)**2 } ; ; b = sqrt{ 340 } = 18.44 ; ;

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{ (4-(-5))**2 + (-5-(-8))**2 } ; ; c = sqrt{ 90 } = 9.49 ; ;


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 = 17.26 ; ; b = 18.44 ; ; c = 9.49 ; ;

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

p = a+b+c = 17.26+18.44+9.49 = 45.19 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45.19 }{ 2 } = 22.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.59 * (22.59-17.26)(22.59-18.44)(22.59-9.49) } ; ; T = sqrt{ 6561 } = 81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 81 }{ 17.26 } = 9.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 81 }{ 18.44 } = 8.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 81 }{ 9.49 } = 17.08 ; ;

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{ 17.26**2-18.44**2-9.49**2 }{ 2 * 18.44 * 9.49 } ) = 67° 50'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18.44**2-17.26**2-9.49**2 }{ 2 * 17.26 * 9.49 } ) = 81° 34'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.49**2-17.26**2-18.44**2 }{ 2 * 18.44 * 17.26 } ) = 30° 35'36" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 81 }{ 22.59 } = 3.58 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17.26 }{ 2 * sin 67° 50'1" } = 9.32 ; ;




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