Triangle calculator

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

You have entered side b, angle α and angle β.

Acute scalene triangle.

Sides: a = 11.04218797561   b = 12.2   c = 11.99442573167

Area: T = 59.41660665835
Perimeter: p = 35.23661370728
Semiperimeter: s = 17.61880685364

Angle ∠ A = α = 54.3° = 54°18' = 0.94877137838 rad
Angle ∠ B = β = 63.8° = 63°48' = 1.11435200628 rad
Angle ∠ C = γ = 61.9° = 61°54' = 1.0880358807 rad

Height: ha = 10.76219477654
Height: hb = 9.74403387842
Height: hc = 9.90774190281

Median: ma = 10.76443080201
Median: mb = 9.78217513035
Median: mc = 9.97107573498

Inradius: r = 3.37224506441
Circumradius: R = 6.79884877112

Vertex coordinates: A[11.99442573167; 0] B[0; 0] C[4.87550545381; 9.90774190281]
Centroid: CG[5.62331039516; 3.30224730094]
Coordinates of the circumscribed circle: U[5.99771286583; 3.20221684863]
Coordinates of the inscribed circle: I[5.41880685364; 3.37224506441]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.7° = 125°42' = 0.94877137838 rad
∠ B' = β' = 116.2° = 116°12' = 1.11435200628 rad
∠ C' = γ' = 118.1° = 118°6' = 1.0880358807 rad

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How did we calculate this triangle?

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

b = 12.2 ; ; alpha = 54.3° ; ; beta = 63.8° ; ;

2. From angle α and angle β we calculate γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 54.3 ° - 63.8 ° = 61.9 ° ; ;

3. From angle α, angle β and side b we calculate a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ b } = fraction{ sin( alpha ) }{ sin ( beta ) } ; ; ; ; a = b * fraction{ sin( alpha ) }{ sin ( beta ) } ; ; ; ; a = 12.2 * fraction{ sin(54° 18') }{ sin (63° 48') } = 11.04 ; ;

4. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 12.2**2+11.04**2 - 2 * 12.2 * 11.04 * cos(61° 54') } ; ; c = 11.99 ; ;


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 = 11.04 ; ; b = 12.2 ; ; c = 11.99 ; ;

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

p = a+b+c = 11.04+12.2+11.99 = 35.24 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35.24 }{ 2 } = 17.62 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.62 * (17.62-11.04)(17.62-12.2)(17.62-11.99) } ; ; T = sqrt{ 3530.27 } = 59.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 59.42 }{ 11.04 } = 10.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 59.42 }{ 12.2 } = 9.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 59.42 }{ 11.99 } = 9.91 ; ;

9. 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{ 11.04**2-12.2**2-11.99**2 }{ 2 * 12.2 * 11.99 } ) = 54° 18' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.2**2-11.04**2-11.99**2 }{ 2 * 11.04 * 11.99 } ) = 63° 48' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.99**2-11.04**2-12.2**2 }{ 2 * 12.2 * 11.04 } ) = 61° 54' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 59.42 }{ 17.62 } = 3.37 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.04 }{ 2 * sin 54° 18' } = 6.8 ; ;




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