Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=39.2; b=28.6; c=14.04110700862 and a=39.2; b=28.6; c=51.18441330889.

#1 Obtuse scalene triangle.

Sides: a = 39.2   b = 28.6   c = 14.04110700862

Area: T = 152.6965946057
Perimeter: p = 81.84110700862
Semiperimeter: s = 40.92105350431

Angle ∠ A = α = 130.493293264° = 130°29'35″ = 2.27875313251 rad
Angle ∠ B = β = 33.7° = 33°42' = 0.58881759579 rad
Angle ∠ C = γ = 15.80770673602° = 15°48'25″ = 0.27658853705 rad

Height: ha = 7.79106094927
Height: hb = 10.67880381858
Height: hc = 21.7549901556

Median: ma = 11.1088367323
Median: mb = 25.73772458624
Median: mc = 33.58658912002

Inradius: r = 3.73215236933
Circumradius: R = 25.7732990216

Vertex coordinates: A[14.04110700862; 0] B[0; 0] C[32.61326015875; 21.7549901556]
Centroid: CG[15.55112238912; 7.25499671853]
Coordinates of the circumscribed circle: U[7.02105350431; 24.79883691476]
Coordinates of the inscribed circle: I[12.32105350431; 3.73215236933]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 49.50770673602° = 49°30'25″ = 2.27875313251 rad
∠ B' = β' = 146.3° = 146°18' = 0.58881759579 rad
∠ C' = γ' = 164.193293264° = 164°11'35″ = 0.27658853705 rad




How did we calculate this triangle?

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

a = 39.2 ; ; b = 28.6 ; ; beta = 33.7° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 28.6**2 = 39.2**2 + c**2 - 2 * 39.2 * c * cos(33° 42') ; ; ; ; ; ; c**2 -65.225c +718.68 =0 ; ; a=1; b=-65.225; c=718.68 ; ; D = b**2 - 4ac = 65.225**2 - 4 * 1 * 718.68 = 1379.60712922 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 65.23 ± sqrt{ 1379.61 } }{ 2 } ; ; c_{1,2} = 32.61260159 ± 18.5715315014 ; ; c_{1} = 51.1841330914 ; ; c_{2} = 14.0410700886 ; ;
 ; ; (c -51.1841330914) (c -14.0410700886) = 0 ; ; ; ; c > 0 ; ; ; ; c = 51.184 ; ;


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 = 39.2 ; ; b = 28.6 ; ; c = 14.04 ; ;

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

p = a+b+c = 39.2+28.6+14.04 = 81.84 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 81.84 }{ 2 } = 40.92 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.92 * (40.92-39.2)(40.92-28.6)(40.92-14.04) } ; ; T = sqrt{ 23316.05 } = 152.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 152.7 }{ 39.2 } = 7.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 152.7 }{ 28.6 } = 10.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 152.7 }{ 14.04 } = 21.75 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 39.2**2-28.6**2-14.04**2 }{ 2 * 28.6 * 14.04 } ) = 130° 29'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.6**2-39.2**2-14.04**2 }{ 2 * 39.2 * 14.04 } ) = 33° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.04**2-39.2**2-28.6**2 }{ 2 * 28.6 * 39.2 } ) = 15° 48'25" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 152.7 }{ 40.92 } = 3.73 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.2 }{ 2 * sin 130° 29'35" } = 25.77 ; ;





#2 Obtuse scalene triangle.

Sides: a = 39.2   b = 28.6   c = 51.18441330889

Area: T = 556.6254927955
Perimeter: p = 118.9844133089
Semiperimeter: s = 59.49220665444

Angle ∠ A = α = 49.50770673602° = 49°30'25″ = 0.86440613284 rad
Angle ∠ B = β = 33.7° = 33°42' = 0.58881759579 rad
Angle ∠ C = γ = 96.79329326398° = 96°47'35″ = 1.68993553672 rad

Height: ha = 28.39992310181
Height: hb = 38.92548201367
Height: hc = 21.7549901556

Median: ma = 36.53439258776
Median: mb = 43.28766924127
Median: mc = 22.85548929113

Inradius: r = 9.35662883303
Circumradius: R = 25.7732990216

Vertex coordinates: A[51.18441330889; 0] B[0; 0] C[32.61326015875; 21.7549901556]
Centroid: CG[27.93222448921; 7.25499671853]
Coordinates of the circumscribed circle: U[25.59220665444; -3.04884675916]
Coordinates of the inscribed circle: I[30.89220665444; 9.35662883303]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.493293264° = 130°29'35″ = 0.86440613284 rad
∠ B' = β' = 146.3° = 146°18' = 0.58881759579 rad
∠ C' = γ' = 83.20770673602° = 83°12'25″ = 1.68993553672 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 39.2 ; ; b = 28.6 ; ; beta = 33.7° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 28.6**2 = 39.2**2 + c**2 - 2 * 39.2 * c * cos(33° 42') ; ; ; ; ; ; c**2 -65.225c +718.68 =0 ; ; a=1; b=-65.225; c=718.68 ; ; D = b**2 - 4ac = 65.225**2 - 4 * 1 * 718.68 = 1379.60712922 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 65.23 ± sqrt{ 1379.61 } }{ 2 } ; ; c_{1,2} = 32.61260159 ± 18.5715315014 ; ; c_{1} = 51.1841330914 ; ; c_{2} = 14.0410700886 ; ; : Nr. 1
 ; ; (c -51.1841330914) (c -14.0410700886) = 0 ; ; ; ; c > 0 ; ; ; ; c = 51.184 ; ; : 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 = 39.2 ; ; b = 28.6 ; ; c = 51.18 ; ;

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

p = a+b+c = 39.2+28.6+51.18 = 118.98 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 118.98 }{ 2 } = 59.49 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.49 * (59.49-39.2)(59.49-28.6)(59.49-51.18) } ; ; T = sqrt{ 309831.31 } = 556.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 556.62 }{ 39.2 } = 28.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 556.62 }{ 28.6 } = 38.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 556.62 }{ 51.18 } = 21.75 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 39.2**2-28.6**2-51.18**2 }{ 2 * 28.6 * 51.18 } ) = 49° 30'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.6**2-39.2**2-51.18**2 }{ 2 * 39.2 * 51.18 } ) = 33° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 51.18**2-39.2**2-28.6**2 }{ 2 * 28.6 * 39.2 } ) = 96° 47'35" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 556.62 }{ 59.49 } = 9.36 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.2 }{ 2 * sin 49° 30'25" } = 25.77 ; ;




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