Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 6.08327625303   b = 11.4021754251   c = 8.54440037453

Area: T = 25.5
Perimeter: p = 26.02985205266
Semiperimeter: s = 13.01442602633

Angle ∠ A = α = 31.56989711293° = 31°34'8″ = 0.55109824877 rad
Angle ∠ B = β = 101.0943723012° = 101°5'37″ = 1.76444183197 rad
Angle ∠ C = γ = 47.33773058591° = 47°20'14″ = 0.82661918463 rad

Height: ha = 8.38443483526
Height: hb = 4.47329958985
Height: hc = 5.9699098507

Median: ma = 9.60546863561
Median: mb = 4.74334164903
Median: mc = 8.07877472107

Inradius: r = 1.95993891227
Circumradius: R = 5.8099433657

Vertex coordinates: A[-9; -5] B[-6; 3] C[0; 2]
Centroid: CG[-5; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.38441939456; 1.95993891227]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.4311028871° = 148°25'52″ = 0.55109824877 rad
∠ B' = β' = 78.90662769884° = 78°54'23″ = 1.76444183197 rad
∠ C' = γ' = 132.6632694141° = 132°39'46″ = 0.82661918463 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{ (-6-0)**2 + (3-2)**2 } ; ; a = sqrt{ 37 } = 6.08 ; ;

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{ (-9-0)**2 + (-5-2)**2 } ; ; b = sqrt{ 130 } = 11.4 ; ;

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{ (-9-(-6))**2 + (-5-3)**2 } ; ; c = sqrt{ 73 } = 8.54 ; ;


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 = 6.08 ; ; b = 11.4 ; ; c = 8.54 ; ;

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

p = a+b+c = 6.08+11.4+8.54 = 26.03 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.03 }{ 2 } = 13.01 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.01 * (13.01-6.08)(13.01-11.4)(13.01-8.54) } ; ; T = sqrt{ 650.25 } = 25.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.5 }{ 6.08 } = 8.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.5 }{ 11.4 } = 4.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.5 }{ 8.54 } = 5.97 ; ;

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{ 6.08**2-11.4**2-8.54**2 }{ 2 * 11.4 * 8.54 } ) = 31° 34'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.4**2-6.08**2-8.54**2 }{ 2 * 6.08 * 8.54 } ) = 101° 5'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.54**2-6.08**2-11.4**2 }{ 2 * 11.4 * 6.08 } ) = 47° 20'14" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.5 }{ 13.01 } = 1.96 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.08 }{ 2 * sin 31° 34'8" } = 5.81 ; ;




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