Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 9.84988578018   b = 13.60114705087   c = 17.72200451467

Area: T = 66.5
Perimeter: p = 41.17703734572
Semiperimeter: s = 20.58551867286

Angle ∠ A = α = 33.49222693031° = 33°29'32″ = 0.58545503733 rad
Angle ∠ B = β = 49.64879706914° = 49°38'53″ = 0.86765205555 rad
Angle ∠ C = γ = 96.86597600055° = 96°51'35″ = 1.69105217248 rad

Height: ha = 13.50441039963
Height: hb = 9.77883544738
Height: hc = 7.50656242182

Median: ma = 15.00883310198
Median: mb = 12.61994294641
Median: mc = 7.90656941504

Inradius: r = 3.23304783472
Circumradius: R = 8.92439045975

Vertex coordinates: A[-10; -3] B[7; 2] C[3; -7]
Centroid: CG[0; -2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.74546921296; 3.23304783472]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.5087730697° = 146°30'28″ = 0.58545503733 rad
∠ B' = β' = 130.3522029309° = 130°21'7″ = 0.86765205555 rad
∠ C' = γ' = 83.14402399945° = 83°8'25″ = 1.69105217248 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{ (7-3)**2 + (2-(-7))**2 } ; ; a = sqrt{ 97 } = 9.85 ; ;

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{ (-10-3)**2 + (-3-(-7))**2 } ; ; b = sqrt{ 185 } = 13.6 ; ;

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{ (-10-7)**2 + (-3-2)**2 } ; ; c = sqrt{ 314 } = 17.72 ; ;


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 = 9.85 ; ; b = 13.6 ; ; c = 17.72 ; ;

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

p = a+b+c = 9.85+13.6+17.72 = 41.17 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41.17 }{ 2 } = 20.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.59 * (20.59-9.85)(20.59-13.6)(20.59-17.72) } ; ; T = sqrt{ 4422.25 } = 66.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66.5 }{ 9.85 } = 13.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66.5 }{ 13.6 } = 9.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66.5 }{ 17.72 } = 7.51 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13.6**2+17.72**2-9.85**2 }{ 2 * 13.6 * 17.72 } ) = 33° 29'32" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.85**2+17.72**2-13.6**2 }{ 2 * 9.85 * 17.72 } ) = 49° 38'53" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 9.85**2+13.6**2-17.72**2 }{ 2 * 9.85 * 13.6 } ) = 96° 51'35" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66.5 }{ 20.59 } = 3.23 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.85 }{ 2 * sin 33° 29'32" } = 8.92 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.6**2+2 * 17.72**2 - 9.85**2 } }{ 2 } = 15.008 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.72**2+2 * 9.85**2 - 13.6**2 } }{ 2 } = 12.619 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.6**2+2 * 9.85**2 - 17.72**2 } }{ 2 } = 7.906 ; ;
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