400 90 450 triangle

Obtuse scalene triangle.

Sides: a = 400   b = 90   c = 450

Area: T = 15812.65331613
Perimeter: p = 940
Semiperimeter: s = 470

Angle ∠ A = α = 51.34404624624° = 51°20'26″ = 0.89660601095 rad
Angle ∠ B = β = 10.11991720527° = 10°7'9″ = 0.17766128699 rad
Angle ∠ C = γ = 118.5440365485° = 118°32'25″ = 2.06989196742 rad

Height: ha = 79.06332658066
Height: hb = 351.3922292474
Height: hc = 70.27884584947

Median: ma = 255.5398646784
Median: mb = 423.3549737215
Median: mc = 182.8255052988

Inradius: r = 33.64439428964
Circumradius: R = 256.1244001373

Vertex coordinates: A[450; 0] B[0; 0] C[393.7787777778; 70.27884584947]
Centroid: CG[281.2599259259; 23.42661528316]
Coordinates of the circumscribed circle: U[225; -122.3770356212]
Coordinates of the inscribed circle: I[380; 33.64439428964]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.6659537538° = 128°39'34″ = 0.89660601095 rad
∠ B' = β' = 169.8810827947° = 169°52'51″ = 0.17766128699 rad
∠ C' = γ' = 61.46596345151° = 61°27'35″ = 2.06989196742 rad

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How did we calculate this triangle?

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 = 400 ; ; b = 90 ; ; c = 450 ; ;

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

p = a+b+c = 400+90+450 = 940 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 940 }{ 2 } = 470 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 470 * (470-400)(470-90)(470-450) } ; ; T = sqrt{ 250040000 } = 15812.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15812.65 }{ 400 } = 79.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15812.65 }{ 90 } = 351.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15812.65 }{ 450 } = 70.28 ; ;

5. 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{ 90**2+450**2-400**2 }{ 2 * 90 * 450 } ) = 51° 20'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 400**2+450**2-90**2 }{ 2 * 400 * 450 } ) = 10° 7'9" ; ; gamma = 180° - alpha - beta = 180° - 51° 20'26" - 10° 7'9" = 118° 32'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15812.65 }{ 470 } = 33.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 400 }{ 2 * sin 51° 20'26" } = 256.12 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 450**2 - 400**2 } }{ 2 } = 255.539 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 450**2+2 * 400**2 - 90**2 } }{ 2 } = 423.35 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 400**2 - 450**2 } }{ 2 } = 182.825 ; ;
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