Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and c.

Acute scalene triangle.

Sides: a = 119   b = 111   c = 90

Area: T = 4743.501081691
Perimeter: p = 320
Semiperimeter: s = 160

Angle ∠ A = α = 71.74109799509° = 71°44'28″ = 1.25221163088 rad
Angle ∠ B = β = 62.35110950415° = 62°21'4″ = 1.08882319007 rad
Angle ∠ C = γ = 45.90879250076° = 45°54'29″ = 0.80112444441 rad

Height: ha = 79.72327028052
Height: hb = 85.46884831875
Height: hc = 105.4111129265

Median: ma = 81.67215984905
Median: mb = 89.72331854093
Median: mc = 105.9065618359

Inradius: r = 29.64768801057
Circumradius: R = 62.65546745688

Vertex coordinates: A[90; 0] B[0; 0] C[55.22222222222; 105.4111129265]
Centroid: CG[48.40774074074; 35.13770430882]
Coordinates of the circumscribed circle: U[45; 43.59659659294]
Coordinates of the inscribed circle: I[49; 29.64768801057]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.2599020049° = 108°15'32″ = 1.25221163088 rad
∠ B' = β' = 117.6498904958° = 117°38'56″ = 1.08882319007 rad
∠ C' = γ' = 134.0922074992° = 134°5'31″ = 0.80112444441 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and c.

a = 119 ; ; b = 111 ; ; c = 90 ; ;


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 = 119 ; ; b = 111 ; ; c = 90 ; ; : Nr. 1

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

p = a+b+c = 119+111+90 = 320 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 320 }{ 2 } = 160 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 160 * (160-119)(160-111)(160-90) } ; ; T = sqrt{ 22500800 } = 4743.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4743.5 }{ 119 } = 79.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4743.5 }{ 111 } = 85.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4743.5 }{ 90 } = 105.41 ; ;

6. 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 111**2+90**2-119**2 }{ 2 * 111 * 90 } ) = 71° 44'28" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 119**2+90**2-111**2 }{ 2 * 119 * 90 } ) = 62° 21'4" ; ; gamma = 180° - alpha - beta = 180° - 71° 44'28" - 62° 21'4" = 45° 54'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4743.5 }{ 160 } = 29.65 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 119 }{ 2 * sin 71° 44'28" } = 62.65 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 111**2+2 * 90**2 - 119**2 } }{ 2 } = 81.672 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 119**2 - 111**2 } }{ 2 } = 89.723 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 111**2+2 * 119**2 - 90**2 } }{ 2 } = 105.906 ; ;
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