Triangle calculator - result

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

You have entered side a, b and angle α.

Right scalene triangle.

Sides: a = 100   b = 40   c = 91.65215138991

Area: T = 1833.033027798
Perimeter: p = 231.6521513899
Semiperimeter: s = 115.826575695

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 23.57881784782° = 23°34'41″ = 0.41215168461 rad
Angle ∠ C = γ = 66.42218215218° = 66°25'19″ = 1.15992794807 rad

Height: ha = 36.66106055596
Height: hb = 91.65215138991
Height: hc = 40

Median: ma = 50
Median: mb = 93.80883151965
Median: mc = 60.8287625303

Inradius: r = 15.82657569496
Circumradius: R = 50

Vertex coordinates: A[91.65215138991; 0] B[0; 0] C[91.65215138991; 40]
Centroid: CG[61.10110092661; 13.33333333333]
Coordinates of the circumscribed circle: U[45.82657569496; 20]
Coordinates of the inscribed circle: I[75.82657569496; 15.82657569496]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 156.4221821522° = 156°25'19″ = 0.41215168461 rad
∠ C' = γ' = 113.5788178478° = 113°34'41″ = 1.15992794807 rad

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How did we calculate this triangle?

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

a = 100 ; ; b = 40 ; ; alpha = 90° ; ;

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 100**2 = 40**2 + c**2 - 2 * 40 * c * cos 90° ; ; ; ; ; ; c**2 -8400 =0 ; ; p=1; q=-0; r=-8400 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-8400) = 33600 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 33600 } }{ 2 } = fraction{ ± 40 sqrt{ 21 } }{ 2 } ; ; c_{1,2} = ± 20 sqrt{ 21} = ± 91.6515138991 ; ; c_{1} = 20 sqrt{ 21} = 91.6515138991 ; ;
c_{2} = - 20 sqrt{ 21} = -91.6515138991 ; ; ; ; text{ Factored form: } ; ; (c -91.6515138991) (c +91.6515138991) = 0 ; ; ; ; c > 0 ; ; ; ; c = 91.652 ; ;
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 = 100 ; ; b = 40 ; ; c = 91.65 ; ;

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

p = a+b+c = 100+40+91.65 = 231.65 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 231.65 }{ 2 } = 115.83 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 115.83 * (115.83-100)(115.83-40)(115.83-91.65) } ; ; T = sqrt{ 3360000 } = 1833.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1833.03 }{ 100 } = 36.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1833.03 }{ 40 } = 91.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1833.03 }{ 91.65 } = 40 ; ;

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{ 40**2+91.65**2-100**2 }{ 2 * 40 * 91.65 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+91.65**2-40**2 }{ 2 * 100 * 91.65 } ) = 23° 34'41" ; ; gamma = 180° - alpha - beta = 180° - 90° - 23° 34'41" = 66° 25'19" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1833.03 }{ 115.83 } = 15.83 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 90° } = 50 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 91.65**2 - 100**2 } }{ 2 } = 50 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 91.65**2+2 * 100**2 - 40**2 } }{ 2 } = 93.808 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 100**2 - 91.65**2 } }{ 2 } = 60.828 ; ;
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