Triangle calculator

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

You have entered side a, b and angle α.

Triangle has two solutions: a=11.4; b=12.8; c=3.33003475024 and a=11.4; b=12.8; c=10.2665585662.

#1 Obtuse scalene triangle.

Sides: a = 11.4   b = 12.8   c = 3.33003475024

Area: T = 17.91326618627
Perimeter: p = 27.55003475024
Semiperimeter: s = 13.75501737512

Angle ∠ A = α = 58° = 1.01222909662 rad
Angle ∠ B = β = 107.7887820505° = 107°47'16″ = 1.88112523614 rad
Angle ∠ C = γ = 14.21221794948° = 14°12'44″ = 0.24880493261 rad

Height: ha = 3.14325722566
Height: hb = 2.7998853416
Height: hc = 10.85550156308

Median: ma = 7.40878436011
Median: mb = 5.4288272913
Median: mc = 12.00773696783

Inradius: r = 1.30327225828
Circumradius: R = 6.72113168992

Vertex coordinates: A[3.33003475024; 0] B[0; 0] C[-3.48326190798; 10.85550156308]
Centroid: CG[-0.06107571925; 3.61883385436]
Coordinates of the circumscribed circle: U[1.65501737512; 6.51655987791]
Coordinates of the inscribed circle: I[0.95501737512; 1.30327225828]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122° = 1.01222909662 rad
∠ B' = β' = 72.21221794948° = 72°12'44″ = 1.88112523614 rad
∠ C' = γ' = 165.7887820505° = 165°47'16″ = 0.24880493261 rad




How did we calculate this triangle?

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

a = 11.4 ; ; b = 12.8 ; ; alpha = 58° ; ;

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 11.4**2 = 12.8**2 + c**2 - 2 * 12.8 * c * cos 58° ; ; ; ; ; ; c**2 -13.566c +33.88 =0 ; ; a=1; b=-13.566; c=33.88 ; ; D = b**2 - 4ac = 13.566**2 - 4 * 1 * 33.88 = 48.5145426202 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 13.57 ± sqrt{ 48.51 } }{ 2 } ; ; c_{1,2} = 6.78296658 ± 3.48261907981 ; ; c_{1} = 10.2655856598 ; ; c_{2} = 3.30034750019 ; ;
 ; ; text{ Factored form: } ; ; (c -10.2655856598) (c -3.30034750019) = 0 ; ; ; ; c > 0 ; ; ; ; c = 10.266 ; ;


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 = 11.4 ; ; b = 12.8 ; ; c = 3.3 ; ;

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

p = a+b+c = 11.4+12.8+3.3 = 27.5 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.5 }{ 2 } = 13.75 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.75 * (13.75-11.4)(13.75-12.8)(13.75-3.3) } ; ; T = sqrt{ 320.86 } = 17.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.91 }{ 11.4 } = 3.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.91 }{ 12.8 } = 2.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.91 }{ 3.3 } = 10.86 ; ;

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{ 12.8**2+3.3**2-11.4**2 }{ 2 * 12.8 * 3.3 } ) = 58° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.4**2+3.3**2-12.8**2 }{ 2 * 11.4 * 3.3 } ) = 107° 47'16" ; ; gamma = 180° - alpha - beta = 180° - 58° - 107° 47'16" = 14° 12'44" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.91 }{ 13.75 } = 1.3 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 11.4 }{ 2 * sin 58° } = 6.72 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.8**2+2 * 3.3**2 - 11.4**2 } }{ 2 } = 7.408 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.3**2+2 * 11.4**2 - 12.8**2 } }{ 2 } = 5.428 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.8**2+2 * 11.4**2 - 3.3**2 } }{ 2 } = 12.007 ; ;







#2 Acute scalene triangle.

Sides: a = 11.4   b = 12.8   c = 10.2665585662

Area: T = 55.71765464101
Perimeter: p = 34.4665585662
Semiperimeter: s = 17.2332792831

Angle ∠ A = α = 58° = 1.01222909662 rad
Angle ∠ B = β = 72.21221794948° = 72°12'44″ = 1.26603402922 rad
Angle ∠ C = γ = 49.78878205052° = 49°47'16″ = 0.86989613952 rad

Height: ha = 9.77548327035
Height: hb = 8.70657103766
Height: hc = 10.85550156308

Median: ma = 10.10554997151
Median: mb = 8.75884887105
Median: mc = 10.98797284918

Inradius: r = 3.23331698615
Circumradius: R = 6.72113168992

Vertex coordinates: A[10.2665585662; 0] B[0; 0] C[3.48326190798; 10.85550156308]
Centroid: CG[4.58327349139; 3.61883385436]
Coordinates of the circumscribed circle: U[5.1332792831; 4.33994168517]
Coordinates of the inscribed circle: I[4.4332792831; 3.23331698615]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122° = 1.01222909662 rad
∠ B' = β' = 107.7887820505° = 107°47'16″ = 1.26603402922 rad
∠ C' = γ' = 130.2122179495° = 130°12'44″ = 0.86989613952 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 11.4 ; ; b = 12.8 ; ; alpha = 58° ; ; : Nr. 1

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 11.4**2 = 12.8**2 + c**2 - 2 * 12.8 * c * cos 58° ; ; ; ; ; ; c**2 -13.566c +33.88 =0 ; ; a=1; b=-13.566; c=33.88 ; ; D = b**2 - 4ac = 13.566**2 - 4 * 1 * 33.88 = 48.5145426202 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 13.57 ± sqrt{ 48.51 } }{ 2 } ; ; c_{1,2} = 6.78296658 ± 3.48261907981 ; ; c_{1} = 10.2655856598 ; ; c_{2} = 3.30034750019 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -10.2655856598) (c -3.30034750019) = 0 ; ; ; ; c > 0 ; ; ; ; c = 10.266 ; ; : 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 = 11.4 ; ; b = 12.8 ; ; c = 10.27 ; ;

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

p = a+b+c = 11.4+12.8+10.27 = 34.47 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34.47 }{ 2 } = 17.23 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.23 * (17.23-11.4)(17.23-12.8)(17.23-10.27) } ; ; T = sqrt{ 3104.33 } = 55.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 55.72 }{ 11.4 } = 9.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 55.72 }{ 12.8 } = 8.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 55.72 }{ 10.27 } = 10.86 ; ;

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{ 12.8**2+10.27**2-11.4**2 }{ 2 * 12.8 * 10.27 } ) = 58° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.4**2+10.27**2-12.8**2 }{ 2 * 11.4 * 10.27 } ) = 72° 12'44" ; ; gamma = 180° - alpha - beta = 180° - 58° - 72° 12'44" = 49° 47'16" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 55.72 }{ 17.23 } = 3.23 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 11.4 }{ 2 * sin 58° } = 6.72 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.8**2+2 * 10.27**2 - 11.4**2 } }{ 2 } = 10.105 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.27**2+2 * 11.4**2 - 12.8**2 } }{ 2 } = 8.758 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.8**2+2 * 11.4**2 - 10.27**2 } }{ 2 } = 10.98 ; ;
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