Triangle calculator

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

You have entered side b, c and angle α.

Acute scalene triangle.

Sides: a = 22.58657330089   b = 31.57   c = 27.66

Area: T = 304.5010660219
Perimeter: p = 81.81657330089
Semiperimeter: s = 40.90878665044

Angle ∠ A = α = 44.22° = 44°13'12″ = 0.77217845952 rad
Angle ∠ B = β = 77.11993911128° = 77°7'10″ = 1.34659872921 rad
Angle ∠ C = γ = 58.66106088872° = 58°39'38″ = 1.02438207663 rad

Height: ha = 26.96439829798
Height: hb = 19.29105074576
Height: hc = 22.01774013173

Median: ma = 27.44770657104
Median: mb = 19.70986083419
Median: mc = 23.7099095676

Inradius: r = 7.44435722573
Circumradius: R = 16.19224557039

Vertex coordinates: A[27.66; 0] B[0; 0] C[5.03548162608; 22.01774013173]
Centroid: CG[10.89882720869; 7.33991337724]
Coordinates of the circumscribed circle: U[13.83; 8.42218003848]
Coordinates of the inscribed circle: I[9.33878665044; 7.44435722573]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.78° = 135°46'48″ = 0.77217845952 rad
∠ B' = β' = 102.8810608887° = 102°52'50″ = 1.34659872921 rad
∠ C' = γ' = 121.3399391113° = 121°20'22″ = 1.02438207663 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 31.57 ; ; c = 27.66 ; ; alpha = 44.22° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 31.57**2+27.66**2 - 2 * 31.57 * 27.66 * cos(44° 13'12") } ; ; a = 22.59 ; ;


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 = 22.59 ; ; b = 31.57 ; ; c = 27.66 ; ;

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

p = a+b+c = 22.59+31.57+27.66 = 81.82 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 81.82 }{ 2 } = 40.91 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.91 * (40.91-22.59)(40.91-31.57)(40.91-27.66) } ; ; T = sqrt{ 92720.65 } = 304.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 304.5 }{ 22.59 } = 26.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 304.5 }{ 31.57 } = 19.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 304.5 }{ 27.66 } = 22.02 ; ;

7. 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{ 31.57**2+27.66**2-22.59**2 }{ 2 * 31.57 * 27.66 } ) = 44° 13'12" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.59**2+27.66**2-31.57**2 }{ 2 * 22.59 * 27.66 } ) = 77° 7'10" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 22.59**2+31.57**2-27.66**2 }{ 2 * 22.59 * 31.57 } ) = 58° 39'38" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 304.5 }{ 40.91 } = 7.44 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22.59 }{ 2 * sin 44° 13'12" } = 16.19 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.57**2+2 * 27.66**2 - 22.59**2 } }{ 2 } = 27.447 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.66**2+2 * 22.59**2 - 31.57**2 } }{ 2 } = 19.709 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.57**2+2 * 22.59**2 - 27.66**2 } }{ 2 } = 23.709 ; ;
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