Triangle calculator

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

You have entered side a, c and angle β.

Obtuse scalene triangle.

Sides: a = 50   b = 50.80329652905   c = 90

Area: T = 1021.479862441
Perimeter: p = 190.803296529
Semiperimeter: s = 95.40114826452

Angle ∠ A = α = 26.54395180817° = 26°32'22″ = 0.46332019724 rad
Angle ∠ B = β = 27° = 0.4711238898 rad
Angle ∠ C = γ = 126.4660481918° = 126°27'38″ = 2.20771517831 rad

Height: ha = 40.85991449766
Height: hb = 40.21333465467
Height: hc = 22.7699524987

Median: ma = 68.6699284554
Median: mb = 68.22658358646
Median: mc = 22.70439785314

Inradius: r = 10.70771567033
Circumradius: R = 55.95215731272

Vertex coordinates: A[90; 0] B[0; 0] C[44.55503262094; 22.7699524987]
Centroid: CG[44.85501087365; 7.5676508329]
Coordinates of the circumscribed circle: U[45; -33.25502411331]
Coordinates of the inscribed circle: I[44.59985173548; 10.70771567033]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4660481918° = 153°27'38″ = 0.46332019724 rad
∠ B' = β' = 153° = 0.4711238898 rad
∠ C' = γ' = 53.54395180817° = 53°32'22″ = 2.20771517831 rad

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How did we calculate this triangle?

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

a = 50 ; ; c = 90 ; ; beta = 27° ; ;

2. 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{ 50**2+90**2 - 2 * 50 * 90 * cos 27° } ; ; b = 50.8 ; ;
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 = 50 ; ; b = 50.8 ; ; c = 90 ; ;

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

p = a+b+c = 50+50.8+90 = 190.8 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 190.8 }{ 2 } = 95.4 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 95.4 * (95.4-50)(95.4-50.8)(95.4-90) } ; ; T = sqrt{ 1043418.58 } = 1021.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1021.48 }{ 50 } = 40.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1021.48 }{ 50.8 } = 40.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1021.48 }{ 90 } = 22.7 ; ;

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{ 50.8**2+90**2-50**2 }{ 2 * 50.8 * 90 } ) = 26° 32'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 50**2+90**2-50.8**2 }{ 2 * 50 * 90 } ) = 27° ; ;
 gamma = 180° - alpha - beta = 180° - 26° 32'22" - 27° = 126° 27'38" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1021.48 }{ 95.4 } = 10.71 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 50 }{ 2 * sin 26° 32'22" } = 55.95 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50.8**2+2 * 90**2 - 50**2 } }{ 2 } = 68.669 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 50**2 - 50.8**2 } }{ 2 } = 68.226 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50.8**2+2 * 50**2 - 90**2 } }{ 2 } = 22.704 ; ;
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