Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=6.3; b=3.30110726037; c=9.3 and a=6.3; b=14.17772101429; c=9.3.

#1 Obtuse scalene triangle.

Sides: a = 6.3   b = 3.30110726037   c = 9.3

Area: T = 5.25500049615
Perimeter: p = 18.90110726037
Semiperimeter: s = 9.45105363019

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 10.32438433776° = 10°19'26″ = 0.18801850584 rad
Angle ∠ C = γ = 149.6766156622° = 149°40'34″ = 2.61223417448 rad

Height: ha = 1.66766682417
Height: hb = 3.18107873329
Height: hc = 1.12990333251

Median: ma = 6.22766395566
Median: mb = 7.77695385909
Median: mc = 1.91659958684

Inradius: r = 0.55655245537
Circumradius: R = 9.21099838605

Vertex coordinates: A[9.3; 0] B[0; 0] C[6.19880064336; 1.12990333251]
Centroid: CG[5.16660021445; 0.37663444417]
Coordinates of the circumscribed circle: U[4.65; -7.95499246984]
Coordinates of the inscribed circle: I[6.14994636981; 0.55655245537]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 169.6766156622° = 169°40'34″ = 0.18801850584 rad
∠ C' = γ' = 30.32438433776° = 30°19'26″ = 2.61223417448 rad




How did we calculate this triangle?

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

a = 6.3 ; ; c = 9.3 ; ; alpha = 20° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 6.3**2 = 9.3**2 + b**2 - 2 * 9.3 * b * cos 20° ; ; ; ; ; ; b**2 -17.478b +46.8 =0 ; ; p=1; q=-17.478; r=46.8 ; ; D = q**2 - 4pr = 17.478**2 - 4 * 1 * 46.8 = 118.290367771 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 17.48 ± sqrt{ 118.29 } }{ 2 } ; ; b_{1,2} = 8.73914137 ± 5.43806876958 ; ; b_{1} = 14.1772101396 ; ; b_{2} = 3.30107260042 ; ; ; ; text{ Factored form: } ; ; (b -14.1772101396) (b -3.30107260042) = 0 ; ; ; ; b > 0 ; ; ; ; b = 14.177 ; ;


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 = 6.3 ; ; b = 3.3 ; ; c = 9.3 ; ;

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

p = a+b+c = 6.3+3.3+9.3 = 18.9 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.9 }{ 2 } = 9.45 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.45 * (9.45-6.3)(9.45-3.3)(9.45-9.3) } ; ; T = sqrt{ 27.56 } = 5.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.25 }{ 6.3 } = 1.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.25 }{ 3.3 } = 3.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.25 }{ 9.3 } = 1.13 ; ;

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{ 3.3**2+9.3**2-6.3**2 }{ 2 * 3.3 * 9.3 } ) = 20° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.3**2+9.3**2-3.3**2 }{ 2 * 6.3 * 9.3 } ) = 10° 19'26" ; ; gamma = 180° - alpha - beta = 180° - 20° - 10° 19'26" = 149° 40'34" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.25 }{ 9.45 } = 0.56 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.3 }{ 2 * sin 20° } = 9.21 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.3**2+2 * 9.3**2 - 6.3**2 } }{ 2 } = 6.227 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 6.3**2 - 3.3**2 } }{ 2 } = 7.77 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.3**2+2 * 6.3**2 - 9.3**2 } }{ 2 } = 1.916 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6.3   b = 14.17772101429   c = 9.3

Area: T = 22.54773452194
Perimeter: p = 29.77772101429
Semiperimeter: s = 14.88986050714

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 129.6766156622° = 129°40'34″ = 2.26332758944 rad
Angle ∠ C = γ = 30.32438433776° = 30°19'26″ = 0.52992509088 rad

Height: ha = 7.15878873712
Height: hb = 3.18107873329
Height: hc = 4.8498891445

Median: ma = 11.56880224636
Median: mb = 3.58435287275
Median: mc = 9.93657507878

Inradius: r = 1.51444028007
Circumradius: R = 9.21099838605

Vertex coordinates: A[9.3; 0] B[0; 0] C[-4.02222197546; 4.8498891445]
Centroid: CG[1.75992600818; 1.61662971483]
Coordinates of the circumscribed circle: U[4.65; 7.95499246984]
Coordinates of the inscribed circle: I[0.71113949286; 1.51444028007]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 50.32438433776° = 50°19'26″ = 2.26332758944 rad
∠ C' = γ' = 149.6766156622° = 149°40'34″ = 0.52992509088 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 6.3 ; ; c = 9.3 ; ; alpha = 20° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 6.3**2 = 9.3**2 + b**2 - 2 * 9.3 * b * cos 20° ; ; ; ; ; ; b**2 -17.478b +46.8 =0 ; ; p=1; q=-17.478; r=46.8 ; ; D = q**2 - 4pr = 17.478**2 - 4 * 1 * 46.8 = 118.290367771 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 17.48 ± sqrt{ 118.29 } }{ 2 } ; ; b_{1,2} = 8.73914137 ± 5.43806876958 ; ; b_{1} = 14.1772101396 ; ; b_{2} = 3.30107260042 ; ; ; ; text{ Factored form: } ; ; (b -14.1772101396) (b -3.30107260042) = 0 ; ; ; ; b > 0 ; ; ; ; b = 14.177 ; ; : 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 = 6.3 ; ; b = 14.18 ; ; c = 9.3 ; ;

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

p = a+b+c = 6.3+14.18+9.3 = 29.78 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.78 }{ 2 } = 14.89 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.89 * (14.89-6.3)(14.89-14.18)(14.89-9.3) } ; ; T = sqrt{ 508.38 } = 22.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.55 }{ 6.3 } = 7.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.55 }{ 14.18 } = 3.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.55 }{ 9.3 } = 4.85 ; ;

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{ 14.18**2+9.3**2-6.3**2 }{ 2 * 14.18 * 9.3 } ) = 20° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.3**2+9.3**2-14.18**2 }{ 2 * 6.3 * 9.3 } ) = 129° 40'34" ; ; gamma = 180° - alpha - beta = 180° - 20° - 129° 40'34" = 30° 19'26" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.55 }{ 14.89 } = 1.51 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.3 }{ 2 * sin 20° } = 9.21 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.18**2+2 * 9.3**2 - 6.3**2 } }{ 2 } = 11.568 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.3**2+2 * 6.3**2 - 14.18**2 } }{ 2 } = 3.584 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.18**2+2 * 6.3**2 - 9.3**2 } }{ 2 } = 9.936 ; ;
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