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=100; b=67; c=54.72441969737 and a=100; b=67; c=100.7054995318.

#1 Obtuse scalene triangle.

Sides: a = 100   b = 67   c = 54.72441969737

Area: T = 1721.953265197
Perimeter: p = 221.7244196974
Semiperimeter: s = 110.8622098487

Angle ∠ A = α = 110.068831319° = 110°4'6″ = 1.92110544673 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 30.93216868105° = 30°55'54″ = 0.5439859778 rad

Height: ha = 34.43990530394
Height: hb = 51.40215717005
Height: hc = 62.9322039105

Median: ma = 35.24401598635
Median: mb = 73.31552021562
Median: mc = 80.59766225496

Inradius: r = 15.53223837044
Circumradius: R = 53.23220269237

Vertex coordinates: A[54.72441969737; 0] B[0; 0] C[77.71545961457; 62.9322039105]
Centroid: CG[44.14662643731; 20.97773463683]
Coordinates of the circumscribed circle: U[27.36220984868; 45.66114088351]
Coordinates of the inscribed circle: I[43.86220984868; 15.53223837044]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 69.93216868105° = 69°55'54″ = 1.92110544673 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 149.068831319° = 149°4'6″ = 0.5439859778 rad




How did we calculate this triangle?

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

a = 100 ; ; b = 67 ; ; beta = 39° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 67**2 = 100**2 + c**2 - 2 * 100 * c * cos 39° ; ; ; ; ; ; c**2 -155.429c +5511 =0 ; ; p=1; q=-155.429; r=5511 ; ; D = q**2 - 4pr = 155.429**2 - 4 * 1 * 5511 = 2114.23381636 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 155.43 ± sqrt{ 2114.23 } }{ 2 } ; ; c_{1,2} = 77.71459615 ± 22.990399172 ; ; c_{1} = 100.704995322 ; ; c_{2} = 54.724196978 ; ; ; ; text{ Factored form: } ; ; (c -100.704995322) (c -54.724196978) = 0 ; ; ; ; c > 0 ; ; ; ; c = 100.705 ; ;


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 = 100 ; ; b = 67 ; ; c = 54.72 ; ;

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

p = a+b+c = 100+67+54.72 = 221.72 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 221.72 }{ 2 } = 110.86 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.86 * (110.86-100)(110.86-67)(110.86-54.72) } ; ; T = sqrt{ 2965120.94 } = 1721.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1721.95 }{ 100 } = 34.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1721.95 }{ 67 } = 51.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1721.95 }{ 54.72 } = 62.93 ; ;

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{ 67**2+54.72**2-100**2 }{ 2 * 67 * 54.72 } ) = 110° 4'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+54.72**2-67**2 }{ 2 * 100 * 54.72 } ) = 39° ; ; gamma = 180° - alpha - beta = 180° - 110° 4'6" - 39° = 30° 55'54" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1721.95 }{ 110.86 } = 15.53 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 110° 4'6" } = 53.23 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 54.72**2 - 100**2 } }{ 2 } = 35.24 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 54.72**2+2 * 100**2 - 67**2 } }{ 2 } = 73.315 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 100**2 - 54.72**2 } }{ 2 } = 80.597 ; ;







#2 Acute scalene triangle.

Sides: a = 100   b = 67   c = 100.7054995318

Area: T = 3168.78553517
Perimeter: p = 267.7054995318
Semiperimeter: s = 133.8522497659

Angle ∠ A = α = 69.93216868105° = 69°55'54″ = 1.22105381863 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 71.06883131895° = 71°4'6″ = 1.2440376059 rad

Height: ha = 63.3765707034
Height: hb = 94.59106075135
Height: hc = 62.9322039105

Median: ma = 69.39219883054
Median: mb = 94.59765012089
Median: mc = 68.62330717727

Inradius: r = 23.67437110411
Circumradius: R = 53.23220269237

Vertex coordinates: A[100.7054995318; 0] B[0; 0] C[77.71545961457; 62.9322039105]
Centroid: CG[59.47331971545; 20.97773463683]
Coordinates of the circumscribed circle: U[50.35224976589; 17.27106302699]
Coordinates of the inscribed circle: I[66.85224976589; 23.67437110411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110.068831319° = 110°4'6″ = 1.22105381863 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 108.932168681° = 108°55'54″ = 1.2440376059 rad

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How did we calculate this triangle?

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

a = 100 ; ; b = 67 ; ; beta = 39° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 67**2 = 100**2 + c**2 - 2 * 100 * c * cos 39° ; ; ; ; ; ; c**2 -155.429c +5511 =0 ; ; p=1; q=-155.429; r=5511 ; ; D = q**2 - 4pr = 155.429**2 - 4 * 1 * 5511 = 2114.23381636 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 155.43 ± sqrt{ 2114.23 } }{ 2 } ; ; c_{1,2} = 77.71459615 ± 22.990399172 ; ; c_{1} = 100.704995322 ; ; c_{2} = 54.724196978 ; ; ; ; text{ Factored form: } ; ; (c -100.704995322) (c -54.724196978) = 0 ; ; ; ; c > 0 ; ; ; ; c = 100.705 ; ; : 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 = 100 ; ; b = 67 ; ; c = 100.7 ; ;

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

p = a+b+c = 100+67+100.7 = 267.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 267.7 }{ 2 } = 133.85 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.85 * (133.85-100)(133.85-67)(133.85-100.7) } ; ; T = sqrt{ 10041200.61 } = 3168.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3168.79 }{ 100 } = 63.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3168.79 }{ 67 } = 94.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3168.79 }{ 100.7 } = 62.93 ; ;

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{ 67**2+100.7**2-100**2 }{ 2 * 67 * 100.7 } ) = 69° 55'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+100.7**2-67**2 }{ 2 * 100 * 100.7 } ) = 39° ; ; gamma = 180° - alpha - beta = 180° - 69° 55'54" - 39° = 71° 4'6" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3168.79 }{ 133.85 } = 23.67 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 69° 55'54" } = 53.23 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 100.7**2 - 100**2 } }{ 2 } = 69.392 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100.7**2+2 * 100**2 - 67**2 } }{ 2 } = 94.597 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 67**2+2 * 100**2 - 100.7**2 } }{ 2 } = 68.623 ; ;
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