Triangle calculator

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

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

Acute scalene triangle.

Sides: a = 105   b = 69.72330878238   c = 113.3232978734

Area: T = 3580.472223121
Perimeter: p = 288.0466066558
Semiperimeter: s = 144.0233033279

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 78° = 1.36113568166 rad

Height: ha = 68.19994710706
Height: hb = 102.7055498077
Height: hc = 63.1910577431

Median: ma = 78.07333837005
Median: mb = 103.5298843858
Median: mc = 68.79441139237

Inradius: r = 24.86604139886
Circumradius: R = 57.92773407455

Vertex coordinates: A[113.3232978734; 0] B[0; 0] C[83.8576728555; 63.1910577431]
Centroid: CG[65.72765690964; 21.06435258103]
Coordinates of the circumscribed circle: U[56.66114893671; 12.0443771359]
Coordinates of the inscribed circle: I[74.32999454552; 24.86604139886]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 102° = 1.36113568166 rad

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How did we calculate this triangle?

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

a = 105 ; ; alpha = 65° ; ; beta = 37° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 65 ° - 37 ° = 78 ° ; ;

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

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 105 * fraction{ sin 37° }{ sin 65° } = 69.72 ; ;

4. From angle γ, angle α and side a we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 105 * fraction{ sin 78° }{ sin 65° } = 113.32 ; ;
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 = 105 ; ; b = 69.72 ; ; c = 113.32 ; ;

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

p = a+b+c = 105+69.72+113.32 = 288.05 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 288.05 }{ 2 } = 144.02 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 144.02 * (144.02-105)(144.02-69.72)(144.02-113.32) } ; ; T = sqrt{ 12819781.4 } = 3580.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3580.47 }{ 105 } = 68.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3580.47 }{ 69.72 } = 102.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3580.47 }{ 113.32 } = 63.19 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 69.72**2+113.32**2-105**2 }{ 2 * 69.72 * 113.32 } ) = 65° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 105**2+113.32**2-69.72**2 }{ 2 * 105 * 113.32 } ) = 37° ; ;
 gamma = 180° - alpha - beta = 180° - 65° - 37° = 78° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3580.47 }{ 144.02 } = 24.86 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 105 }{ 2 * sin 65° } = 57.93 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 69.72**2+2 * 113.32**2 - 105**2 } }{ 2 } = 78.073 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 113.32**2+2 * 105**2 - 69.72**2 } }{ 2 } = 103.529 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 69.72**2+2 * 105**2 - 113.32**2 } }{ 2 } = 68.794 ; ;
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