Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse isosceles triangle.

Sides: a = 12.72879220614   b = 6.70882039325   c = 6.70882039325

Area: T = 13.5
Perimeter: p = 26.14443299264
Semiperimeter: s = 13.07221649632

Angle ∠ A = α = 143.1330102354° = 143°7'48″ = 2.49880915448 rad
Angle ∠ B = β = 18.43549488229° = 18°26'6″ = 0.32217505544 rad
Angle ∠ C = γ = 18.43549488229° = 18°26'6″ = 0.32217505544 rad

Height: ha = 2.12113203436
Height: hb = 4.02549223595
Height: hc = 4.02549223595

Median: ma = 2.12113203436
Median: mb = 9.60546863561
Median: mc = 9.60546863561

Inradius: r = 1.03327287055
Circumradius: R = 10.60766017178

Vertex coordinates: A[-2; -1] B[1; 5] C[-8; -4]
Centroid: CG[-3; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.09881861164; 1.03327287055]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 36.87698976458° = 36°52'12″ = 2.49880915448 rad
∠ B' = β' = 161.5655051177° = 161°33'54″ = 0.32217505544 rad
∠ C' = γ' = 161.5655051177° = 161°33'54″ = 0.32217505544 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-(-8))**2 + (5-(-4))**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{ (-2-(-8))**2 + (-1-(-4))**2 } ; ; b = sqrt{ 45 } = 6.71 ; ;

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{ (-2-1)**2 + (-1-5)**2 } ; ; c = sqrt{ 45 } = 6.71 ; ;


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 = 6.71 ; ; c = 6.71 ; ;

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

p = a+b+c = 12.73+6.71+6.71 = 26.14 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.14 }{ 2 } = 13.07 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.07 * (13.07-12.73)(13.07-6.71)(13.07-6.71) } ; ; T = sqrt{ 182.25 } = 13.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.5 }{ 12.73 } = 2.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.5 }{ 6.71 } = 4.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.5 }{ 6.71 } = 4.02 ; ;

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**2-6.71**2-6.71**2 }{ 2 * 6.71 * 6.71 } ) = 143° 7'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.71**2-12.73**2-6.71**2 }{ 2 * 12.73 * 6.71 } ) = 18° 26'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.71**2-12.73**2-6.71**2 }{ 2 * 6.71 * 12.73 } ) = 18° 26'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.5 }{ 13.07 } = 1.03 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.73 }{ 2 * sin 143° 7'48" } = 10.61 ; ;




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