Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 11.4021754251   b = 9.43439811321   c = 7.81102496759

Area: T = 36.5
Perimeter: p = 28.6465985059
Semiperimeter: s = 14.32329925295

Angle ∠ A = α = 82.21998121158° = 82°11'59″ = 1.43546573659 rad
Angle ∠ B = β = 55.06106897953° = 55°3'38″ = 0.96109903253 rad
Angle ∠ C = γ = 42.73994980889° = 42°44'22″ = 0.74659449623 rad

Height: ha = 6.40325235409
Height: hb = 7.7387984524
Height: hc = 9.34766922351

Median: ma = 6.51992024052
Median: mb = 8.55986213843
Median: mc = 9.70882439195

Inradius: r = 2.54883501388
Circumradius: R = 5.75441177012

Vertex coordinates: A[4; 0] B[-1; 6] C[-4; -5]
Centroid: CG[-0.33333333333; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.78803542065; 2.54883501388]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.88001878842° = 97°48'1″ = 1.43546573659 rad
∠ B' = β' = 124.9399310205° = 124°56'22″ = 0.96109903253 rad
∠ C' = γ' = 137.2610501911° = 137°15'38″ = 0.74659449623 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{ (-1-(-4))**2 + (6-(-5))**2 } ; ; a = sqrt{ 130 } = 11.4 ; ;

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-(-4))**2 + (0-(-5))**2 } ; ; b = sqrt{ 89 } = 9.43 ; ;

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-(-1))**2 + (0-6)**2 } ; ; c = sqrt{ 61 } = 7.81 ; ;


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 = 11.4 ; ; b = 9.43 ; ; c = 7.81 ; ;

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

p = a+b+c = 11.4+9.43+7.81 = 28.65 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28.65 }{ 2 } = 14.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.32 * (14.32-11.4)(14.32-9.43)(14.32-7.81) } ; ; T = sqrt{ 1332.25 } = 36.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36.5 }{ 11.4 } = 6.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36.5 }{ 9.43 } = 7.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36.5 }{ 7.81 } = 9.35 ; ;

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{ 11.4**2-9.43**2-7.81**2 }{ 2 * 9.43 * 7.81 } ) = 82° 11'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.43**2-11.4**2-7.81**2 }{ 2 * 11.4 * 7.81 } ) = 55° 3'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.81**2-11.4**2-9.43**2 }{ 2 * 9.43 * 11.4 } ) = 42° 44'22" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36.5 }{ 14.32 } = 2.55 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.4 }{ 2 * sin 82° 11'59" } = 5.75 ; ;




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