Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 11.70546999107   b = 6.70882039325   c = 5.09990195136

Area: T = 4.5
Perimeter: p = 23.51219233568
Semiperimeter: s = 11.75659616784

Angle ∠ A = α = 164.7454881297° = 164°44'42″ = 2.87553406044 rad
Angle ∠ B = β = 8.67331740479° = 8°40'23″ = 0.15113754437 rad
Angle ∠ C = γ = 6.58219446552° = 6°34'55″ = 0.11548766054 rad

Height: ha = 0.76989218919
Height: hb = 1.34216407865
Height: hc = 1.76550452162

Median: ma = 1.11880339887
Median: mb = 8.38215273071
Median: mc = 9.19223881554

Inradius: r = 0.38327845074
Circumradius: R = 22.24223519939

Vertex coordinates: A[-7; 3] B[-8; -2] C[-4; 9]
Centroid: CG[-6.33333333333; 3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.5099365104; 0.38327845074]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 15.25551187031° = 15°15'18″ = 2.87553406044 rad
∠ B' = β' = 171.3276825952° = 171°19'37″ = 0.15113754437 rad
∠ C' = γ' = 173.4188055345° = 173°25'5″ = 0.11548766054 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-(-4))**2 + (-2-9)**2 } ; ; a = sqrt{ 137 } = 11.7 ; ;

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{ (-7-(-4))**2 + (3-9)**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{ (-7-(-8))**2 + (3-(-2))**2 } ; ; c = sqrt{ 26 } = 5.1 ; ;


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 = 11.7 ; ; b = 6.71 ; ; c = 5.1 ; ;

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

p = a+b+c = 11.7+6.71+5.1 = 23.51 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.51 }{ 2 } = 11.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.76 * (11.76-11.7)(11.76-6.71)(11.76-5.1) } ; ; T = sqrt{ 20.25 } = 4.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.5 }{ 11.7 } = 0.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.5 }{ 6.71 } = 1.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.5 }{ 5.1 } = 1.77 ; ;

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{ 11.7**2-6.71**2-5.1**2 }{ 2 * 6.71 * 5.1 } ) = 164° 44'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.71**2-11.7**2-5.1**2 }{ 2 * 11.7 * 5.1 } ) = 8° 40'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.1**2-11.7**2-6.71**2 }{ 2 * 6.71 * 11.7 } ) = 6° 34'55" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.5 }{ 11.76 } = 0.38 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.7 }{ 2 * sin 164° 44'42" } = 22.24 ; ;




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