Triangle calculator

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

You have entered side a, c and angle β.

Obtuse scalene triangle.

Sides: a = 12.7   b = 9.31986327517   c = 5.8

Area: T = 25.0243823236
Perimeter: p = 27.81986327517
Semiperimeter: s = 13.90993163759

Angle ∠ A = α = 112.1832799719° = 112°10'58″ = 1.95879592192 rad
Angle ∠ B = β = 42.8° = 42°48' = 0.74770009199 rad
Angle ∠ C = γ = 25.01772002808° = 25°1'2″ = 0.43766325145 rad

Height: ha = 3.94107595647
Height: hb = 5.37107070345
Height: hc = 8.62989045641

Median: ma = 4.46327299023
Median: mb = 8.70437791165
Median: mc = 10.75442297809

Inradius: r = 1.79990692396
Circumradius: R = 6.85875701045

Vertex coordinates: A[5.8; 0] B[0; 0] C[9.31883692792; 8.62989045641]
Centroid: CG[5.03994564264; 2.87663015214]
Coordinates of the circumscribed circle: U[2.9; 6.21441988814]
Coordinates of the inscribed circle: I[4.59106836241; 1.79990692396]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.81772002808° = 67°49'2″ = 1.95879592192 rad
∠ B' = β' = 137.2° = 137°12' = 0.74770009199 rad
∠ C' = γ' = 154.9832799719° = 154°58'58″ = 0.43766325145 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 12.7 ; ; c = 5.8 ; ; beta = 42.8° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 12.7**2+5.8**2 - 2 * 12.7 * 5.8 * cos(42° 48') } ; ; b = 9.32 ; ;


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 = 12.7 ; ; b = 9.32 ; ; c = 5.8 ; ;

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

p = a+b+c = 12.7+9.32+5.8 = 27.82 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.82 }{ 2 } = 13.91 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.91 * (13.91-12.7)(13.91-9.32)(13.91-5.8) } ; ; T = sqrt{ 626.19 } = 25.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.02 }{ 12.7 } = 3.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.02 }{ 9.32 } = 5.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.02 }{ 5.8 } = 8.63 ; ;

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{ 9.32**2+5.8**2-12.7**2 }{ 2 * 9.32 * 5.8 } ) = 112° 10'58" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.7**2+5.8**2-9.32**2 }{ 2 * 12.7 * 5.8 } ) = 42° 48' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 12.7**2+9.32**2-5.8**2 }{ 2 * 12.7 * 9.32 } ) = 25° 1'2" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.02 }{ 13.91 } = 1.8 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.7 }{ 2 * sin 112° 10'58" } = 6.86 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.32**2+2 * 5.8**2 - 12.7**2 } }{ 2 } = 4.463 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.8**2+2 * 12.7**2 - 9.32**2 } }{ 2 } = 8.704 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.32**2+2 * 12.7**2 - 5.8**2 } }{ 2 } = 10.754 ; ;
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