Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 16.55329453572   b = 19.02662975904   c = 7.21111025509

Area: T = 59
Perimeter: p = 42.79903454986
Semiperimeter: s = 21.39551727493

Angle ∠ A = α = 59.32327199782° = 59°19'22″ = 1.03553767849 rad
Angle ∠ B = β = 98.67331740479° = 98°40'23″ = 1.72221717705 rad
Angle ∠ C = γ = 22.00441059739° = 22°15″ = 0.38440440982 rad

Height: ha = 7.12986407013
Height: hb = 6.20219423085
Height: hc = 16.36436557886

Median: ma = 11.76986022959
Median: mb = 8.5154693183
Median: mc = 17.46442491966

Inradius: r = 2.75876313915
Circumradius: R = 9.62331938765

Vertex coordinates: A[-9; -6] B[-5; 0] C[10; -7]
Centroid: CG[-1.33333333333; -4.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.4210655636; 2.75876313915]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.6777280022° = 120°40'38″ = 1.03553767849 rad
∠ B' = β' = 81.32768259521° = 81°19'37″ = 1.72221717705 rad
∠ C' = γ' = 157.9965894026° = 157°59'45″ = 0.38440440982 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{ (-5-10)**2 + (0-(-7))**2 } ; ; a = sqrt{ 274 } = 16.55 ; ;

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{ (-9-10)**2 + (-6-(-7))**2 } ; ; b = sqrt{ 362 } = 19.03 ; ;

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{ (-9-(-5))**2 + (-6-0)**2 } ; ; c = sqrt{ 52 } = 7.21 ; ;


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 = 16.55 ; ; b = 19.03 ; ; c = 7.21 ; ;

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

p = a+b+c = 16.55+19.03+7.21 = 42.79 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42.79 }{ 2 } = 21.4 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.4 * (21.4-16.55)(21.4-19.03)(21.4-7.21) } ; ; T = sqrt{ 3481 } = 59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 59 }{ 16.55 } = 7.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 59 }{ 19.03 } = 6.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 59 }{ 7.21 } = 16.36 ; ;

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{ 16.55**2-19.03**2-7.21**2 }{ 2 * 19.03 * 7.21 } ) = 59° 19'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19.03**2-16.55**2-7.21**2 }{ 2 * 16.55 * 7.21 } ) = 98° 40'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.21**2-16.55**2-19.03**2 }{ 2 * 19.03 * 16.55 } ) = 22° 15" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 59 }{ 21.4 } = 2.76 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.55 }{ 2 * sin 59° 19'22" } = 9.62 ; ;




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