Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 7.07110678119   b = 15.81113883008   c = 14.14221356237

Area: T = 50
Perimeter: p = 37.02545917364
Semiperimeter: s = 18.51222958682

Angle ∠ A = α = 26.56550511771° = 26°33'54″ = 0.4643647609 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 63.43549488229° = 63°26'6″ = 1.10771487178 rad

Height: ha = 14.14221356237
Height: hb = 6.32545553203
Height: hc = 7.07110678119

Median: ma = 14.57773797371
Median: mb = 7.90656941504
Median: mc = 10

Inradius: r = 2.70109075674
Circumradius: R = 7.90656941504

Vertex coordinates: A[5; 15] B[15; 5] C[20; 10]
Centroid: CG[13.33333333333; 10]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.70109075674]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 116.5655051177° = 116°33'54″ = 1.10771487178 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{ (15-20)**2 + (5-10)**2 } ; ; a = sqrt{ 50 } = 7.07 ; ;

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{ (5-20)**2 + (15-10)**2 } ; ; b = sqrt{ 250 } = 15.81 ; ;

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{ (5-15)**2 + (15-5)**2 } ; ; c = sqrt{ 200 } = 14.14 ; ;


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 = 7.07 ; ; b = 15.81 ; ; c = 14.14 ; ;

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

p = a+b+c = 7.07+15.81+14.14 = 37.02 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.02 }{ 2 } = 18.51 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.51 * (18.51-7.07)(18.51-15.81)(18.51-14.14) } ; ; T = sqrt{ 2500 } = 50 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50 }{ 7.07 } = 14.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50 }{ 15.81 } = 6.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50 }{ 14.14 } = 7.07 ; ;

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{ 7.07**2-15.81**2-14.14**2 }{ 2 * 15.81 * 14.14 } ) = 26° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.81**2-7.07**2-14.14**2 }{ 2 * 7.07 * 14.14 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.14**2-7.07**2-15.81**2 }{ 2 * 15.81 * 7.07 } ) = 63° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50 }{ 18.51 } = 2.7 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.07 }{ 2 * sin 26° 33'54" } = 7.91 ; ;




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