Triangle calculator

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

You have entered side a, b and angle α.

Obtuse scalene triangle.

Sides: a = 37   b = 34   c = 3.09768396657

Area: T = 13.62658584471
Perimeter: p = 74.09768396657
Semiperimeter: s = 37.04884198329

Angle ∠ A = α = 165° = 2.88797932658 rad
Angle ∠ B = β = 13.75987197856° = 13°45'31″ = 0.24401349611 rad
Angle ∠ C = γ = 1.24112802144° = 1°14'29″ = 0.02216644267 rad

Height: ha = 0.7376532889
Height: hb = 0.80215210851
Height: hc = 8.87998475335

Median: ma = 15.51095199138
Median: mb = 20.00773788378
Median: mc = 35.49879210098

Inradius: r = 0.36877851446
Circumradius: R = 71.47985111454

Vertex coordinates: A[3.09768396657; 0] B[0; 0] C[35.93883177596; 8.87998475335]
Centroid: CG[13.01217191418; 2.93332825112]
Coordinates of the circumscribed circle: U[1.54884198329; 71.46217376754]
Coordinates of the inscribed circle: I[3.04884198329; 0.36877851446]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 15° = 2.88797932658 rad
∠ B' = β' = 166.2411280214° = 166°14'29″ = 0.24401349611 rad
∠ C' = γ' = 178.7598719786° = 178°45'31″ = 0.02216644267 rad

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How did we calculate this triangle?

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

a = 37 ; ; b = 34 ; ; alpha = 165° ; ;

2. From angle α, b and side a we calculate c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 37**2 = 34**2 + c**2 - 2 * 34 * c * cos(165° ) ; ; ; ; ; ; c**2 +65.683c -213 =0 ; ; a=1; b=65.683; c=-213 ; ; D = b**2 - 4ac = 65.683**2 - 4 * 1 * (-213) = 5166.25073355 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ -65.68 ± sqrt{ 5166.25 } }{ 2 } ; ; c_{1,2} = -32.84147809 ± 35.9383177596 ; ; c_{1} = 3.09683966956 ; ; c_{2} = -68.7797958496 ; ;
 ; ; (c -3.09683966956) (c +68.7797958496) = 0 ; ; ; ; c > 0 ; ; ; ; c = 3.097 ; ;


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 = 37 ; ; b = 34 ; ; c = 3.1 ; ;

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

p = a+b+c = 37+34+3.1 = 74.1 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74.1 }{ 2 } = 37.05 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.05 * (37.05-37)(37.05-34)(37.05-3.1) } ; ; T = sqrt{ 185.66 } = 13.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.63 }{ 37 } = 0.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.63 }{ 34 } = 0.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.63 }{ 3.1 } = 8.8 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37**2-34**2-3.1**2 }{ 2 * 34 * 3.1 } ) = 165° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34**2-37**2-3.1**2 }{ 2 * 37 * 3.1 } ) = 13° 45'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.1**2-37**2-34**2 }{ 2 * 34 * 37 } ) = 1° 14'29" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.63 }{ 37.05 } = 0.37 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37 }{ 2 * sin 165° } = 71.48 ; ;




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