Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 8.6022325267   b = 8.24662112512   c = 7.07110678119

Area: T = 27
Perimeter: p = 23.92196043301
Semiperimeter: s = 11.96598021651

Angle ∠ A = α = 67.83436541779° = 67°50'1″ = 1.18439206091 rad
Angle ∠ B = β = 62.59224245622° = 62°35'33″ = 1.09224438954 rad
Angle ∠ C = γ = 49.57439212599° = 49°34'26″ = 0.86552281491 rad

Height: ha = 6.27773724922
Height: hb = 6.5488461876
Height: hc = 7.63767532368

Median: ma = 6.36439610307
Median: mb = 6.70882039325
Median: mc = 7.64985292704

Inradius: r = 2.25875624268
Circumradius: R = 4.64444208163

Vertex coordinates: A[0; -10] B[1; -3] C[8; -8]
Centroid: CG[3; -7]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.1710587925; 2.25875624268]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.1666345822° = 112°9'59″ = 1.18439206091 rad
∠ B' = β' = 117.4087575438° = 117°24'27″ = 1.09224438954 rad
∠ C' = γ' = 130.426607874° = 130°25'34″ = 0.86552281491 rad

Calculate another triangle




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{ (1-8)**2 + (-3-(-8))**2 } ; ; a = sqrt{ 74 } = 8.6 ; ;

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{ (0-8)**2 + (-10-(-8))**2 } ; ; b = sqrt{ 68 } = 8.25 ; ;

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{ (0-1)**2 + (-10-(-3))**2 } ; ; c = sqrt{ 50 } = 7.07 ; ;


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 = 8.6 ; ; b = 8.25 ; ; c = 7.07 ; ;

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

p = a+b+c = 8.6+8.25+7.07 = 23.92 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.92 }{ 2 } = 11.96 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.96 * (11.96-8.6)(11.96-8.25)(11.96-7.07) } ; ; T = sqrt{ 729 } = 27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27 }{ 8.6 } = 6.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27 }{ 8.25 } = 6.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27 }{ 7.07 } = 7.64 ; ;

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{ 8.6**2-8.25**2-7.07**2 }{ 2 * 8.25 * 7.07 } ) = 67° 50'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.25**2-8.6**2-7.07**2 }{ 2 * 8.6 * 7.07 } ) = 62° 35'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.07**2-8.6**2-8.25**2 }{ 2 * 8.25 * 8.6 } ) = 49° 34'26" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27 }{ 11.96 } = 2.26 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.6 }{ 2 * sin 67° 50'1" } = 4.64 ; ;




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