Triangle calculator

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

You have entered side c, angle α and angle γ.

Obtuse scalene triangle.

Sides: a = 16.7755349753   b = 32.36106439249   c = 37.5

Area: T = 270.7365910315
Perimeter: p = 86.6365993678
Semiperimeter: s = 43.3187996839

Angle ∠ A = α = 26.5° = 26°30' = 0.46325122518 rad
Angle ∠ B = β = 59.4° = 59°24' = 1.03767255757 rad
Angle ∠ C = γ = 94.1° = 94°6' = 1.64223548261 rad

Height: ha = 32.27878260126
Height: hb = 16.73224179916
Height: hc = 14.43992485501

Median: ma = 34.006555172
Median: mb = 24.12552639542
Median: mc = 17.68547198815

Inradius: r = 6.25499637581
Circumradius: R = 18.798810844

Vertex coordinates: A[37.5; 0] B[0; 0] C[8.5399347788; 14.43992485501]
Centroid: CG[15.34664492627; 4.813308285]
Coordinates of the circumscribed circle: U[18.75; -1.34440167117]
Coordinates of the inscribed circle: I[10.9577352914; 6.25499637581]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.5° = 153°30' = 0.46325122518 rad
∠ B' = β' = 120.6° = 120°36' = 1.03767255757 rad
∠ C' = γ' = 85.9° = 85°54' = 1.64223548261 rad

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How did we calculate this triangle?

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

c = 37.5 ; ; alpha = 26.5° ; ; gamma = 94.1° ; ;

2. From angle α, angle γ and side c we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 37.5 * fraction{ sin 26° 30' }{ sin 94° 6' } = 16.78 ; ;

3. 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{ 16.78**2+37.5**2 - 2 * 16.78 * 37.5 * cos 59° 24' } ; ; b = 32.36 ; ;
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 = 16.78 ; ; b = 32.36 ; ; c = 37.5 ; ;

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

p = a+b+c = 16.78+32.36+37.5 = 86.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 86.64 }{ 2 } = 43.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.32 * (43.32-16.78)(43.32-32.36)(43.32-37.5) } ; ; T = sqrt{ 73297.93 } = 270.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 270.74 }{ 16.78 } = 32.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 270.74 }{ 32.36 } = 16.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 270.74 }{ 37.5 } = 14.44 ; ;

8. 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{ 32.36**2+37.5**2-16.78**2 }{ 2 * 32.36 * 37.5 } ) = 26° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16.78**2+37.5**2-32.36**2 }{ 2 * 16.78 * 37.5 } ) = 59° 24' ; ; gamma = 180° - alpha - beta = 180° - 26° 30' - 59° 24' = 94° 6' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 270.74 }{ 43.32 } = 6.25 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16.78 }{ 2 * sin 26° 30' } = 18.8 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.36**2+2 * 37.5**2 - 16.78**2 } }{ 2 } = 34.006 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.5**2+2 * 16.78**2 - 32.36**2 } }{ 2 } = 24.125 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.36**2+2 * 16.78**2 - 37.5**2 } }{ 2 } = 17.685 ; ;
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