Triangle calculator

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

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

Acute scalene triangle.

Sides: a = 34.10987752861   b = 24.76221084339   c = 40

Area: T = 419.9899178284
Perimeter: p = 98.87108837201
Semiperimeter: s = 49.435544186

Angle ∠ A = α = 58° = 1.01222909662 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 84° = 1.46660765717 rad

Height: ha = 24.6266459013
Height: hb = 33.92219238463
Height: hc = 20.99994589142

Median: ma = 28.56109675811
Median: mb = 35.04987342459
Median: mc = 22.09771781639

Inradius: r = 8.49657100105
Circumradius: R = 20.11101655913

Vertex coordinates: A[40; 0] B[0; 0] C[26.87880817178; 20.99994589142]
Centroid: CG[22.29326939059; 76.9998196381]
Coordinates of the circumscribed circle: U[20; 2.10220847053]
Coordinates of the inscribed circle: I[24.67333334261; 8.49657100105]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122° = 1.01222909662 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 96° = 1.46660765717 rad

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How did we calculate this triangle?

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

c = 40 ; ; alpha = 58° ; ; beta = 38° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 58 ° - 38 ° = 84 ° ; ;

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

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 40 * fraction{ sin 58° }{ sin 84° } = 34.11 ; ;

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

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 34.11**2+40**2 - 2 * 34.11 * 40 * cos 38° } ; ; b = 24.76 ; ;


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 = 34.11 ; ; b = 24.76 ; ; c = 40 ; ;

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

p = a+b+c = 34.11+24.76+40 = 98.87 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 98.87 }{ 2 } = 49.44 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 49.44 * (49.44-34.11)(49.44-24.76)(49.44-40) } ; ; T = sqrt{ 176390.91 } = 419.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 419.99 }{ 34.11 } = 24.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 419.99 }{ 24.76 } = 33.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 419.99 }{ 40 } = 21 ; ;

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{ 24.76**2+40**2-34.11**2 }{ 2 * 24.76 * 40 } ) = 58° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 34.11**2+40**2-24.76**2 }{ 2 * 34.11 * 40 } ) = 38° ; ; gamma = 180° - alpha - beta = 180° - 58° - 38° = 84° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 419.99 }{ 49.44 } = 8.5 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 34.11 }{ 2 * sin 58° } = 20.11 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 24.76**2+2 * 40**2 - 34.11**2 } }{ 2 } = 28.561 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 34.11**2 - 24.76**2 } }{ 2 } = 35.049 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 24.76**2+2 * 34.11**2 - 40**2 } }{ 2 } = 22.097 ; ;
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