Triangle calculator

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

You have entered side b, c and angle α.

Obtuse scalene triangle.

Sides: a = 55.89657458458   b = 111.8   c = 125

Area: T = 3124.572217246
Perimeter: p = 292.6965745846
Semiperimeter: s = 146.3487872923

Angle ∠ A = α = 26.562° = 26°33'43″ = 0.46435943559 rad
Angle ∠ B = β = 63.4311464653° = 63°25'53″ = 1.10770879076 rad
Angle ∠ C = γ = 90.0076535347° = 90°24″ = 1.57109103901 rad

Height: ha = 111.8799999273
Height: hb = 55.89657454822
Height: hc = 49.99331547593

Median: ma = 115.2433378982
Median: mb = 79.05660383641
Median: mc = 62.49442973545

Inradius: r = 21.35503080711
Circumradius: R = 62.55000004066

Vertex coordinates: A[125; 0] B[0; 0] C[255.0003776146; 49.99331547593]
Centroid: CG[500.0001258715; 16.66443849198]
Coordinates of the circumscribed circle: U[62.5; -0.00771289577]
Coordinates of the inscribed circle: I[34.54878729229; 21.35503080711]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.438° = 153°26'17″ = 0.46435943559 rad
∠ B' = β' = 116.5698535347° = 116°34'7″ = 1.10770879076 rad
∠ C' = γ' = 89.9933464653° = 89°59'36″ = 1.57109103901 rad

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How did we calculate this triangle?

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

b = 111.8 ; ; c = 125 ; ; alpha = 26.562° ; ;

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{ 111.8**2+125**2 - 2 * 111.8 * 125 * cos(26° 33'43") } ; ; a = 55.9 ; ;


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 = 55.9 ; ; b = 111.8 ; ; c = 125 ; ;

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

p = a+b+c = 55.9+111.8+125 = 292.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 292.7 }{ 2 } = 146.35 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 146.35 * (146.35-55.9)(146.35-111.8)(146.35-125) } ; ; T = sqrt{ 9762951.26 } = 3124.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3124.57 }{ 55.9 } = 111.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3124.57 }{ 111.8 } = 55.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3124.57 }{ 125 } = 49.99 ; ;

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{ 111.8**2+125**2-55.9**2 }{ 2 * 111.8 * 125 } ) = 26° 33'43" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 55.9**2+125**2-111.8**2 }{ 2 * 55.9 * 125 } ) = 63° 25'53" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 55.9**2+111.8**2-125**2 }{ 2 * 55.9 * 111.8 } ) = 90° 24" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3124.57 }{ 146.35 } = 21.35 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 55.9 }{ 2 * sin 26° 33'43" } = 62.5 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.8**2+2 * 125**2 - 55.9**2 } }{ 2 } = 115.243 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 125**2+2 * 55.9**2 - 111.8**2 } }{ 2 } = 79.056 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.8**2+2 * 55.9**2 - 125**2 } }{ 2 } = 62.494 ; ;
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