Triangle calculator

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

You have entered side b, c and angle β.

Triangle has two solutions: a=3.86994850299; b=8.1; c=10.6 and a=12.08217110387; b=8.1; c=10.6.

#1 Obtuse scalene triangle.

Sides: a = 3.86994850299   b = 8.1   c = 10.6

Area: T = 13.50985817281
Perimeter: p = 22.56994850299
Semiperimeter: s = 11.2854742515

Angle ∠ A = α = 18.341061001° = 18°20'26″ = 0.32201040315 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 120.459938999° = 120°27'34″ = 2.10224129703 rad

Height: ha = 6.98221082773
Height: hb = 3.33554522785
Height: hc = 2.54987890053

Median: ma = 9.23326470419
Median: mb = 6.87548787043
Median: mc = 3.49330584304

Inradius: r = 1.19770660128
Circumradius: R = 6.1498572651

Vertex coordinates: A[10.6; 0] B[0; 0] C[2.91114582263; 2.54987890053]
Centroid: CG[4.50438194088; 0.85495963351]
Coordinates of the circumscribed circle: U[5.3; -3.11768807556]
Coordinates of the inscribed circle: I[3.1854742515; 1.19770660128]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.659938999° = 161°39'34″ = 0.32201040315 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 59.541061001° = 59°32'26″ = 2.10224129703 rad




How did we calculate this triangle?

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

b = 8.1 ; ; c = 10.6 ; ; beta = 41.2° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 8.1**2 = 10.6**2 + a**2 - 2 * 10.6 * a * cos 41° 12' ; ; ; ; ; ; a**2 -15.951a +46.75 =0 ; ; p=1; q=-15.951; r=46.75 ; ; D = q**2 - 4pr = 15.951**2 - 4 * 1 * 46.75 = 67.4406560186 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.95 ± sqrt{ 67.44 } }{ 2 } ; ; a_{1,2} = 7.97559803 ± 4.10611300437 ; ; a_{1} = 12.0817110344 ; ; a_{2} = 3.86948502563 ; ; ; ; text{ Factored form: } ; ; (a -12.0817110344) (a -3.86948502563) = 0 ; ; ; ; a > 0 ; ; ; ; a = 12.082 ; ;


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 = 3.87 ; ; b = 8.1 ; ; c = 10.6 ; ;

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

p = a+b+c = 3.87+8.1+10.6 = 22.57 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.57 }{ 2 } = 11.28 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.28 * (11.28-3.87)(11.28-8.1)(11.28-10.6) } ; ; T = sqrt{ 182.48 } = 13.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.51 }{ 3.87 } = 6.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.51 }{ 8.1 } = 3.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.51 }{ 10.6 } = 2.55 ; ;

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{ 8.1**2+10.6**2-3.87**2 }{ 2 * 8.1 * 10.6 } ) = 18° 20'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.87**2+10.6**2-8.1**2 }{ 2 * 3.87 * 10.6 } ) = 41° 12' ; ; gamma = 180° - alpha - beta = 180° - 18° 20'26" - 41° 12' = 120° 27'34" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.51 }{ 11.28 } = 1.2 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.87 }{ 2 * sin 18° 20'26" } = 6.15 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 10.6**2 - 3.87**2 } }{ 2 } = 9.233 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.6**2+2 * 3.87**2 - 8.1**2 } }{ 2 } = 6.875 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 3.87**2 - 10.6**2 } }{ 2 } = 3.493 ; ;







#2 Acute scalene triangle.

Sides: a = 12.08217110387   b = 8.1   c = 10.6

Area: T = 42.17879073232
Perimeter: p = 30.78217110387
Semiperimeter: s = 15.39108555193

Angle ∠ A = α = 79.259938999° = 79°15'34″ = 1.38333373184 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 59.541061001° = 59°32'26″ = 1.03991796833 rad

Height: ha = 6.98221082773
Height: hb = 10.41442981045
Height: hc = 7.95880957214

Median: ma = 7.24552097688
Median: mb = 10.61989157079
Median: mc = 8.81546962971

Inradius: r = 2.74404524245
Circumradius: R = 6.1498572651

Vertex coordinates: A[10.6; 0] B[0; 0] C[9.09904595105; 7.95880957214]
Centroid: CG[6.56334865035; 2.65326985738]
Coordinates of the circumscribed circle: U[5.3; 3.11768807556]
Coordinates of the inscribed circle: I[7.29108555193; 2.74404524245]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 100.741061001° = 100°44'26″ = 1.38333373184 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 120.459938999° = 120°27'34″ = 1.03991796833 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 8.1 ; ; c = 10.6 ; ; beta = 41.2° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 8.1**2 = 10.6**2 + a**2 - 2 * 10.6 * a * cos 41° 12' ; ; ; ; ; ; a**2 -15.951a +46.75 =0 ; ; p=1; q=-15.951; r=46.75 ; ; D = q**2 - 4pr = 15.951**2 - 4 * 1 * 46.75 = 67.4406560186 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.95 ± sqrt{ 67.44 } }{ 2 } ; ; a_{1,2} = 7.97559803 ± 4.10611300437 ; ; a_{1} = 12.0817110344 ; ; a_{2} = 3.86948502563 ; ; ; ; text{ Factored form: } ; ; (a -12.0817110344) (a -3.86948502563) = 0 ; ; ; ; a > 0 ; ; ; ; a = 12.082 ; ; : 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 = 12.08 ; ; b = 8.1 ; ; c = 10.6 ; ;

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

p = a+b+c = 12.08+8.1+10.6 = 30.78 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.78 }{ 2 } = 15.39 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.39 * (15.39-12.08)(15.39-8.1)(15.39-10.6) } ; ; T = sqrt{ 1778.98 } = 42.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.18 }{ 12.08 } = 6.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.18 }{ 8.1 } = 10.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.18 }{ 10.6 } = 7.96 ; ;

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{ 8.1**2+10.6**2-12.08**2 }{ 2 * 8.1 * 10.6 } ) = 79° 15'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.08**2+10.6**2-8.1**2 }{ 2 * 12.08 * 10.6 } ) = 41° 12' ; ; gamma = 180° - alpha - beta = 180° - 79° 15'34" - 41° 12' = 59° 32'26" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.18 }{ 15.39 } = 2.74 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.08 }{ 2 * sin 79° 15'34" } = 6.15 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 10.6**2 - 12.08**2 } }{ 2 } = 7.245 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.6**2+2 * 12.08**2 - 8.1**2 } }{ 2 } = 10.619 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 12.08**2 - 10.6**2 } }{ 2 } = 8.815 ; ;
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