1523 411.5 1540 triangle

Acute scalene triangle.

Sides: a = 1523   b = 411.5   c = 1540

Area: T = 311982.978803
Perimeter: p = 3474.5
Semiperimeter: s = 1737.25

Angle ∠ A = α = 79.93994893515° = 79°56'22″ = 1.39552072915 rad
Angle ∠ B = β = 15.42985161254° = 15°25'43″ = 0.26992784051 rad
Angle ∠ C = γ = 84.63219945231° = 84°37'55″ = 1.4777106957 rad

Height: ha = 409.6955309298
Height: hb = 1516.321067086
Height: hc = 405.173269874

Median: ma = 831.0143763424
Median: mb = 1517.644008826
Median: mc = 807.1744469988

Inradius: r = 179.5844387987
Circumradius: R = 773.3921817795

Vertex coordinates: A[1540; 0] B[0; 0] C[1468.116582792; 405.173269874]
Centroid: CG[1002.705527597; 135.0587566247]
Coordinates of the circumscribed circle: U[770; 72.35326352865]
Coordinates of the inscribed circle: I[1325.75; 179.5844387987]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 100.0610510649° = 100°3'38″ = 1.39552072915 rad
∠ B' = β' = 164.5711483875° = 164°34'17″ = 0.26992784051 rad
∠ C' = γ' = 95.36880054769° = 95°22'5″ = 1.4777106957 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 = 1523 ; ; b = 411.5 ; ; c = 1540 ; ;

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

p = a+b+c = 1523+411.5+1540 = 3474.5 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3474.5 }{ 2 } = 1737.25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1737.25 * (1737.25-1523)(1737.25-411.5)(1737.25-1540) } ; ; T = sqrt{ 97333378580.6 } = 311982.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 311982.98 }{ 1523 } = 409.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 311982.98 }{ 411.5 } = 1516.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 311982.98 }{ 1540 } = 405.17 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1523**2-411.5**2-1540**2 }{ 2 * 411.5 * 1540 } ) = 79° 56'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 411.5**2-1523**2-1540**2 }{ 2 * 1523 * 1540 } ) = 15° 25'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1540**2-1523**2-411.5**2 }{ 2 * 411.5 * 1523 } ) = 84° 37'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 311982.98 }{ 1737.25 } = 179.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1523 }{ 2 * sin 79° 56'22" } = 773.39 ; ;




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