Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 9.48768329805   b = 3.60655512755   c = 12.04215945788

Area: T = 13.5
Perimeter: p = 25.13439788348
Semiperimeter: s = 12.56769894174

Angle ∠ A = α = 38.45437092167° = 38°27'13″ = 0.67111438354 rad
Angle ∠ B = β = 13.67113071322° = 13°40'17″ = 0.23986093225 rad
Angle ∠ C = γ = 127.8754983651° = 127°52'30″ = 2.23218394956 rad

Height: ha = 2.84660498942
Height: hb = 7.4888452649
Height: hc = 2.24222279561

Median: ma = 7.51766481892
Median: mb = 10.68987791632
Median: mc = 3.9055124838

Inradius: r = 1.07442429672
Circumradius: R = 7.62875167876

Vertex coordinates: A[1; -5] B[2; 7] C[-1; -2]
Centroid: CG[0.66766666667; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[4.41663321983; 1.07442429672]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.5466290783° = 141°32'47″ = 0.67111438354 rad
∠ B' = β' = 166.3298692868° = 166°19'43″ = 0.23986093225 rad
∠ C' = γ' = 52.12550163489° = 52°7'30″ = 2.23218394956 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-(-1))**2 + (7-(-2))**2 } ; ; a = sqrt{ 90 } = 9.49 ; ;

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{ (1-(-1))**2 + (-5-(-2))**2 } ; ; b = sqrt{ 13 } = 3.61 ; ;

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{ (1-2)**2 + (-5-7)**2 } ; ; c = sqrt{ 145 } = 12.04 ; ;


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 = 9.49 ; ; b = 3.61 ; ; c = 12.04 ; ;

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

p = a+b+c = 9.49+3.61+12.04 = 25.13 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.13 }{ 2 } = 12.57 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.57 * (12.57-9.49)(12.57-3.61)(12.57-12.04) } ; ; 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 }{ 9.49 } = 2.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.5 }{ 3.61 } = 7.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.5 }{ 12.04 } = 2.24 ; ;

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{ 9.49**2-3.61**2-12.04**2 }{ 2 * 3.61 * 12.04 } ) = 38° 27'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.61**2-9.49**2-12.04**2 }{ 2 * 9.49 * 12.04 } ) = 13° 40'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.04**2-9.49**2-3.61**2 }{ 2 * 3.61 * 9.49 } ) = 127° 52'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.5 }{ 12.57 } = 1.07 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.49 }{ 2 * sin 38° 27'13" } = 7.63 ; ;




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