Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse isosceles triangle.

Sides: a = 12.72879220614   b = 8.06222577483   c = 8.06222577483

Area: T = 31.5
Perimeter: p = 28.8522437558
Semiperimeter: s = 14.4266218779

Angle ∠ A = α = 104.2550032698° = 104°15' = 1.82195063159 rad
Angle ∠ B = β = 37.87549836511° = 37°52'30″ = 0.66110431689 rad
Angle ∠ C = γ = 37.87549836511° = 37°52'30″ = 0.66110431689 rad

Height: ha = 4.95497474683
Height: hb = 7.81441882791
Height: hc = 7.81441882791

Median: ma = 4.95497474683
Median: mb = 9.86215414617
Median: mc = 9.86215414617

Inradius: r = 2.18435243512
Circumradius: R = 6.56659915396

Vertex coordinates: A[6; 5] B[-2; 6] C[7; -3]
Centroid: CG[3.66766666667; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.80773884516; 2.18435243512]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 75.75499673022° = 75°45' = 1.82195063159 rad
∠ B' = β' = 142.1255016349° = 142°7'30″ = 0.66110431689 rad
∠ C' = γ' = 142.1255016349° = 142°7'30″ = 0.66110431689 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{ (-2-7)**2 + (6-(-3))**2 } ; ; a = sqrt{ 162 } = 12.73 ; ;

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{ (6-7)**2 + (5-(-3))**2 } ; ; b = sqrt{ 65 } = 8.06 ; ;

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{ (6-(-2))**2 + (5-6)**2 } ; ; c = sqrt{ 65 } = 8.06 ; ;


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 = 12.73 ; ; b = 8.06 ; ; c = 8.06 ; ;

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

p = a+b+c = 12.73+8.06+8.06 = 28.85 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28.85 }{ 2 } = 14.43 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.43 * (14.43-12.73)(14.43-8.06)(14.43-8.06) } ; ; T = sqrt{ 992.25 } = 31.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.5 }{ 12.73 } = 4.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.5 }{ 8.06 } = 7.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.5 }{ 8.06 } = 7.81 ; ;

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{ 12.73**2-8.06**2-8.06**2 }{ 2 * 8.06 * 8.06 } ) = 104° 15' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.06**2-12.73**2-8.06**2 }{ 2 * 12.73 * 8.06 } ) = 37° 52'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.06**2-12.73**2-8.06**2 }{ 2 * 8.06 * 12.73 } ) = 37° 52'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.5 }{ 14.43 } = 2.18 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.73 }{ 2 * sin 104° 15' } = 6.57 ; ;




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