Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 15.03332963784   b = 5.38551648071   c = 10.05498756211

Area: T = 12.5
Perimeter: p = 30.46883368066
Semiperimeter: s = 15.23441684033

Angle ∠ A = α = 152.4887997376° = 152°29'17″ = 2.6611417624 rad
Angle ∠ B = β = 9.52546679718° = 9°31'29″ = 0.16662368163 rad
Angle ∠ C = γ = 17.98773346521° = 17°59'14″ = 0.31439382133 rad

Height: ha = 1.66329752631
Height: hb = 4.64223834544
Height: hc = 2.48875929755

Median: ma = 2.91554759474
Median: mb = 12.5
Median: mc = 10.11218742081

Inradius: r = 0.82105239478
Circumradius: R = 16.27221111107

Vertex coordinates: A[2; -5] B[-8; -4] C[7; -3]
Centroid: CG[0.33333333333; -4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[4.8990322729; 0.82105239478]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 27.51220026239° = 27°30'43″ = 2.6611417624 rad
∠ B' = β' = 170.4755332028° = 170°28'31″ = 0.16662368163 rad
∠ C' = γ' = 162.0132665348° = 162°46″ = 0.31439382133 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{ (-8-7)**2 + (-4-(-3))**2 } ; ; a = sqrt{ 226 } = 15.03 ; ;

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{ (2-7)**2 + (-5-(-3))**2 } ; ; b = sqrt{ 29 } = 5.39 ; ;

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-(-8))**2 + (-5-(-4))**2 } ; ; c = sqrt{ 101 } = 10.05 ; ;


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 = 15.03 ; ; b = 5.39 ; ; c = 10.05 ; ;

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

p = a+b+c = 15.03+5.39+10.05 = 30.47 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.47 }{ 2 } = 15.23 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.23 * (15.23-15.03)(15.23-5.39)(15.23-10.05) } ; ; T = sqrt{ 156.25 } = 12.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.5 }{ 15.03 } = 1.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.5 }{ 5.39 } = 4.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.5 }{ 10.05 } = 2.49 ; ;

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{ 15.03**2-5.39**2-10.05**2 }{ 2 * 5.39 * 10.05 } ) = 152° 29'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.39**2-15.03**2-10.05**2 }{ 2 * 15.03 * 10.05 } ) = 9° 31'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.05**2-15.03**2-5.39**2 }{ 2 * 5.39 * 15.03 } ) = 17° 59'14" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.5 }{ 15.23 } = 0.82 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15.03 }{ 2 * sin 152° 29'17" } = 16.27 ; ;




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