Triangle calculator

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

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 145.8155375047   b = 111.1   c = 94.44

Area: T = 5246.142
Perimeter: p = 351.3555375047
Semiperimeter: s = 175.6787687523

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 49.63439664686° = 49°38'2″ = 0.86662761357 rad
Angle ∠ C = γ = 40.36660335314° = 40°21'58″ = 0.70545201911 rad

Height: ha = 71.95659511241
Height: hb = 94.44
Height: hc = 111.1

Median: ma = 72.90876875233
Median: mb = 109.5666035339
Median: mc = 120.71884261

Inradius: r = 29.86223124767
Circumradius: R = 72.90876875233

Vertex coordinates: A[94.44; 0] B[0; 0] C[94.44; 111.1]
Centroid: CG[62.96; 37.03333333333]
Coordinates of the circumscribed circle: U[47.22; 55.55]
Coordinates of the inscribed circle: I[64.57876875233; 29.86223124767]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 130.3666033531° = 130°21'58″ = 0.86662761357 rad
∠ C' = γ' = 139.6343966469° = 139°38'2″ = 0.70545201911 rad

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How did we calculate this triangle?

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

b = 111.1 ; ; c = 94.44 ; ; alpha = 90° ; ;

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{ 111.1**2+94.44**2 - 2 * 111.1 * 94.44 * cos 90° } ; ; a = 145.82 ; ;
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 = 145.82 ; ; b = 111.1 ; ; c = 94.44 ; ;

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

p = a+b+c = 145.82+111.1+94.44 = 351.36 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 351.36 }{ 2 } = 175.68 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 175.68 * (175.68-145.82)(175.68-111.1)(175.68-94.44) } ; ; T = sqrt{ 27522005.88 } = 5246.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5246.14 }{ 145.82 } = 71.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5246.14 }{ 111.1 } = 94.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5246.14 }{ 94.44 } = 111.1 ; ;

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{ 111.1**2+94.44**2-145.82**2 }{ 2 * 111.1 * 94.44 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 145.82**2+94.44**2-111.1**2 }{ 2 * 145.82 * 94.44 } ) = 49° 38'2" ; ; gamma = 180° - alpha - beta = 180° - 90° - 49° 38'2" = 40° 21'58" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5246.14 }{ 175.68 } = 29.86 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 145.82 }{ 2 * sin 90° } = 72.91 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.1**2+2 * 94.44**2 - 145.82**2 } }{ 2 } = 72.908 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 94.44**2+2 * 145.82**2 - 111.1**2 } }{ 2 } = 109.566 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.1**2+2 * 145.82**2 - 94.44**2 } }{ 2 } = 120.718 ; ;
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