Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 11.3143708499   b = 12.16655250606   c = 7.21111025509

Area: T = 40
Perimeter: p = 30.69903361105
Semiperimeter: s = 15.34551680553

Angle ∠ A = α = 65.7722254682° = 65°46'20″ = 1.14879424007 rad
Angle ∠ B = β = 78.6990067526° = 78°41'24″ = 1.37334007669 rad
Angle ∠ C = γ = 35.5387677792° = 35°32'16″ = 0.6220249486 rad

Height: ha = 7.07110678119
Height: hb = 6.57659594922
Height: hc = 11.09440039245

Median: ma = 8.24662112512
Median: mb = 7.28801098893
Median: mc = 11.18803398875

Inradius: r = 2.60766837363
Circumradius: R = 6.20332249677

Vertex coordinates: A[7; 4] B[3; -2] C[-5; 6]
Centroid: CG[1.66766666667; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.52113367473; 2.60766837363]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.2287745318° = 114°13'40″ = 1.14879424007 rad
∠ B' = β' = 101.3109932474° = 101°18'36″ = 1.37334007669 rad
∠ C' = γ' = 144.4622322208° = 144°27'44″ = 0.6220249486 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{ (3-(-5))**2 + (-2-6)**2 } ; ; a = sqrt{ 128 } = 11.31 ; ;

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{ (7-(-5))**2 + (4-6)**2 } ; ; b = sqrt{ 148 } = 12.17 ; ;

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{ (7-3)**2 + (4-(-2))**2 } ; ; c = sqrt{ 52 } = 7.21 ; ;


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 = 11.31 ; ; b = 12.17 ; ; c = 7.21 ; ;

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

p = a+b+c = 11.31+12.17+7.21 = 30.69 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.69 }{ 2 } = 15.35 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.35 * (15.35-11.31)(15.35-12.17)(15.35-7.21) } ; ; T = sqrt{ 1600 } = 40 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40 }{ 11.31 } = 7.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40 }{ 12.17 } = 6.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40 }{ 7.21 } = 11.09 ; ;

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{ 11.31**2-12.17**2-7.21**2 }{ 2 * 12.17 * 7.21 } ) = 65° 46'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.17**2-11.31**2-7.21**2 }{ 2 * 11.31 * 7.21 } ) = 78° 41'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.21**2-11.31**2-12.17**2 }{ 2 * 12.17 * 11.31 } ) = 35° 32'16" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40 }{ 15.35 } = 2.61 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.31 }{ 2 * sin 65° 46'20" } = 6.2 ; ;




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