Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 11.4021754251   b = 5.83109518948   c = 12.16655250606

Area: T = 33
Perimeter: p = 29.39882312064
Semiperimeter: s = 14.69991156032

Angle ∠ A = α = 68.4998565676° = 68°29'55″ = 1.19655255039 rad
Angle ∠ B = β = 28.41326614431° = 28°24'46″ = 0.49658944914 rad
Angle ∠ C = γ = 83.0898772881° = 83°5'20″ = 1.45501726582 rad

Height: ha = 5.78985829274
Height: hb = 11.31989066194
Height: hc = 5.42551665811

Median: ma = 7.64985292704
Median: mb = 11.42436596588
Median: mc = 6.70882039325

Inradius: r = 2.24550330272
Circumradius: R = 6.12772847166

Vertex coordinates: A[-4; 2] B[8; 4] C[-1; -3]
Centroid: CG[1; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[4.15499095351; 2.24550330272]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.5011434324° = 111°30'5″ = 1.19655255039 rad
∠ B' = β' = 151.5877338557° = 151°35'14″ = 0.49658944914 rad
∠ C' = γ' = 96.9111227119° = 96°54'40″ = 1.45501726582 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-(-1))**2 + (4-(-3))**2 } ; ; a = sqrt{ 130 } = 11.4 ; ;

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{ (-4-(-1))**2 + (2-(-3))**2 } ; ; b = sqrt{ 34 } = 5.83 ; ;

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{ (-4-8)**2 + (2-4)**2 } ; ; c = sqrt{ 148 } = 12.17 ; ;


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 = 11.4 ; ; b = 5.83 ; ; c = 12.17 ; ;

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

p = a+b+c = 11.4+5.83+12.17 = 29.4 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.4 }{ 2 } = 14.7 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.7 * (14.7-11.4)(14.7-5.83)(14.7-12.17) } ; ; T = sqrt{ 1089 } = 33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 33 }{ 11.4 } = 5.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 33 }{ 5.83 } = 11.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 33 }{ 12.17 } = 5.43 ; ;

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{ 11.4**2-5.83**2-12.17**2 }{ 2 * 5.83 * 12.17 } ) = 68° 29'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.83**2-11.4**2-12.17**2 }{ 2 * 11.4 * 12.17 } ) = 28° 24'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.17**2-11.4**2-5.83**2 }{ 2 * 5.83 * 11.4 } ) = 83° 5'20" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 33 }{ 14.7 } = 2.25 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.4 }{ 2 * sin 68° 29'55" } = 6.13 ; ;




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