Triangle calculator

You have entered side a, c and angle γ.

Triangle has two solutions: a=18.7; b=3.31112488771; c=16.1 and a=18.7; b=27.32550375794; c=16.1.

#1 Obtuse scalene triangle.

Sides: a = 18.7 b = 3.31112488771 c = 16.1

Area: T = 17.75880279927
Perimeter: p = 38.11112488771
Semiperimeter: s = 19.05656244385

Angle ∠ A = α = 138.2255264641° = 138°13'31″ = 2.41224859774 rad
Angle ∠ B = β = 6.77547353587° = 6°46'29″ = 0.1188241438 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: h_a = 1.89992543308
Height: h_b = 10.72658793598
Height: h_c = 2.20659662103

Median: m_a = 6.90439615123
Median: m_b = 17.37697699386
Median: m_c = 10.74882409985

Inradius: r = 0.93219048058
Circumradius: R = 14.03547467047

Vertex coordinates: A[16.1; 0] B[0; 0] C[18.56994295303; 2.20659662103]
Centroid: CG[11.55664765101; 0.73553220701]
Coordinates of the circumscribed circle: U[8.05; 11.49765914543]
Coordinates of the inscribed circle: I[15.74443755615; 0.93219048058]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.77547353587° = 41°46'29″ = 2.41224859774 rad
∠ B' = β' = 173.2255264641° = 173°13'31″ = 0.1188241438 rad
∠ C' = γ' = 145° = 0.61108652382 rad

How did we calculate this triangle?

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

$a = 18.7 ; ; c = 16.1 ; ; gamma = 35° ; ;$

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

$c**2 = a**2 + b**2 - 2a b cos gamma ; ; ; ; 16.1**2 = 18.7**2 + b**2 - 2 * 18.7 * b * cos 35° ; ; ; ; ; ; b**2 -30.636b +90.48 =0 ; ; p=1; q=-30.636; r=90.48 ; ; D = q**2 - 4pr = 30.636**2 - 4 * 1 * 90.48 = 576.662047839 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.64 ± sqrt{ 576.66 } }{ 2 } ; ; b_{1,2} = 15.31814323 ± 12.0068943512 ; ; b_{1} = 27.3250375812 ; ; b_{2} = 3.31124887885 ; ; ; ; text{ Factored form: } ; ; (b -27.3250375812) (b -3.31124887885) = 0 ; ; ; ; b > 0 ; ; ; ; b = 27.325 ; ;$

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.

$a = 18.7 ; ; b = 3.31 ; ; c = 16.1 ; ;$

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

$p = a+b+c = 18.7+3.31+16.1 = 38.11 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 38.11 }{ 2 } = 19.06 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.06 * (19.06-18.7)(19.06-3.31)(19.06-16.1) } ; ; T = sqrt{ 315.35 } = 17.76 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.76 }{ 18.7 } = 1.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.76 }{ 3.31 } = 10.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.76 }{ 16.1 } = 2.21 ; ;$

7. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 3.31**2+16.1**2-18.7**2 }{ 2 * 3.31 * 16.1 } ) = 138° 13'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.7**2+16.1**2-3.31**2 }{ 2 * 18.7 * 16.1 } ) = 6° 46'29" ; ; gamma = 180° - alpha - beta = 180° - 138° 13'31" - 6° 46'29" = 35° ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.76 }{ 19.06 } = 0.93 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.7 }{ 2 * sin 138° 13'31" } = 14.03 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.31**2+2 * 16.1**2 - 18.7**2 } }{ 2 } = 6.904 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.1**2+2 * 18.7**2 - 3.31**2 } }{ 2 } = 17.37 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.31**2+2 * 18.7**2 - 16.1**2 } }{ 2 } = 10.748 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 18.7 b = 27.32550375794 c = 16.1

Area: T = 146.5432528289
Perimeter: p = 62.12550375794
Semiperimeter: s = 31.06325187897

Angle ∠ A = α = 41.77547353587° = 41°46'29″ = 0.72991066762 rad
Angle ∠ B = β = 103.2255264641° = 103°13'31″ = 1.80216207392 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: h_a = 15.67329976779
Height: h_b = 10.72658793598
Height: h_c = 18.20440407812

Median: m_a = 20.38440952548
Median: m_b = 10.8532906538
Median: m_c = 21.98657076156

Inradius: r = 4.7187664053
Circumradius: R = 14.03547467047

Vertex coordinates: A[16.1; 0] B[0; 0] C[-4.27881887799; 18.20440407812]
Centroid: CG[3.941060374; 6.06880135937]
Coordinates of the circumscribed circle: U[8.05; 11.49765914543]
Coordinates of the inscribed circle: I[3.73774812103; 4.7187664053]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2255264641° = 138°13'31″ = 0.72991066762 rad
∠ B' = β' = 76.77547353587° = 76°46'29″ = 1.80216207392 rad
∠ C' = γ' = 145° = 0.61108652382 rad

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How did we calculate this triangle?

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

$a = 18.7 ; ; c = 16.1 ; ; gamma = 35° ; ; : Nr. 1$

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

$c**2 = a**2 + b**2 - 2a b cos gamma ; ; ; ; 16.1**2 = 18.7**2 + b**2 - 2 * 18.7 * b * cos 35° ; ; ; ; ; ; b**2 -30.636b +90.48 =0 ; ; p=1; q=-30.636; r=90.48 ; ; D = q**2 - 4pr = 30.636**2 - 4 * 1 * 90.48 = 576.662047839 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.64 ± sqrt{ 576.66 } }{ 2 } ; ; b_{1,2} = 15.31814323 ± 12.0068943512 ; ; b_{1} = 27.3250375812 ; ; b_{2} = 3.31124887885 ; ; ; ; text{ Factored form: } ; ; (b -27.3250375812) (b -3.31124887885) = 0 ; ; ; ; b > 0 ; ; ; ; b = 27.325 ; ; : Nr. 1$

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.

$a = 18.7 ; ; b = 27.33 ; ; c = 16.1 ; ;$

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

$p = a+b+c = 18.7+27.33+16.1 = 62.13 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 62.13 }{ 2 } = 31.06 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.06 * (31.06-18.7)(31.06-27.33)(31.06-16.1) } ; ; T = sqrt{ 21474.71 } = 146.54 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 146.54 }{ 18.7 } = 15.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 146.54 }{ 27.33 } = 10.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 146.54 }{ 16.1 } = 18.2 ; ;$

7. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 27.33**2+16.1**2-18.7**2 }{ 2 * 27.33 * 16.1 } ) = 41° 46'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.7**2+16.1**2-27.33**2 }{ 2 * 18.7 * 16.1 } ) = 103° 13'31" ; ; gamma = 180° - alpha - beta = 180° - 41° 46'29" - 103° 13'31" = 35° ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 146.54 }{ 31.06 } = 4.72 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.7 }{ 2 * sin 41° 46'29" } = 14.03 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.33**2+2 * 16.1**2 - 18.7**2 } }{ 2 } = 20.384 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.1**2+2 * 18.7**2 - 27.33**2 } }{ 2 } = 10.853 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.33**2+2 * 18.7**2 - 16.1**2 } }{ 2 } = 21.986 ; ;$
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