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 = 350.1911487603   b = 408   c = 90

Area: T = 12982.48105026
Perimeter: p = 848.1911487603
Semiperimeter: s = 424.0965743802

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 124.5329570942° = 124°31'46″ = 2.1733451029 rad
Angle ∠ C = γ = 10.47704290582° = 10°28'14″ = 0.18327434612 rad

Height: ha = 74.14550375704
Height: hb = 63.64396103068
Height: hc = 288.5499566724

Median: ma = 237.9576887907
Median: mb = 154.1143720982
Median: mc = 377.5243560847

Inradius: r = 30.61221452345
Circumradius: R = 247.6232775598

Vertex coordinates: A[90; 0] B[0; 0] C[-198.5499566724; 288.5499566724]
Centroid: CG[-36.16765222414; 96.16765222414]
Coordinates of the circumscribed circle: U[45; 243.5499566724]
Coordinates of the inscribed circle: I[16.09657438015; 30.61221452345]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 55.47704290582° = 55°28'14″ = 2.1733451029 rad
∠ C' = γ' = 169.5329570942° = 169°31'46″ = 0.18327434612 rad

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How did we calculate this triangle?

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

b = 408 ; ; c = 90 ; ; alpha = 45° ; ;

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{ 408**2+90**2 - 2 * 408 * 90 * cos 45° } ; ; a = 350.19 ; ;
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 = 350.19 ; ; b = 408 ; ; c = 90 ; ;

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

p = a+b+c = 350.19+408+90 = 848.19 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 848.19 }{ 2 } = 424.1 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 424.1 * (424.1-350.19)(424.1-408)(424.1-90) } ; ; T = sqrt{ 168544800 } = 12982.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12982.48 }{ 350.19 } = 74.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12982.48 }{ 408 } = 63.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12982.48 }{ 90 } = 288.5 ; ;

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{ 408**2+90**2-350.19**2 }{ 2 * 408 * 90 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 350.19**2+90**2-408**2 }{ 2 * 350.19 * 90 } ) = 124° 31'46" ; ; gamma = 180° - alpha - beta = 180° - 45° - 124° 31'46" = 10° 28'14" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12982.48 }{ 424.1 } = 30.61 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 350.19 }{ 2 * sin 45° } = 247.62 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 408**2+2 * 90**2 - 350.19**2 } }{ 2 } = 237.957 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 350.19**2 - 408**2 } }{ 2 } = 154.114 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 408**2+2 * 350.19**2 - 90**2 } }{ 2 } = 377.524 ; ;
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