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=15.8; b=43.72; c=40.17768308898 and a=15.8; b=43.72; c=41.3622107543.

#1 Obtuse scalene triangle.

Sides: a = 15.8   b = 43.72   c = 40.17768308898

Area: T = 317.1743611487
Perimeter: p = 99.69768308898
Semiperimeter: s = 49.84884154449

Angle ∠ A = α = 21.17° = 21°10'12″ = 0.36994862026 rad
Angle ∠ B = β = 92.15495975958° = 92°8'59″ = 1.60883138824 rad
Angle ∠ C = γ = 66.68804024042° = 66°40'49″ = 1.16437925685 rad

Height: ha = 40.1498558416
Height: hb = 14.50993143407
Height: hc = 15.78988815251

Median: ma = 41.23658832835
Median: mb = 21.30884319032
Median: mc = 26.0199123062

Inradius: r = 6.36327621592
Circumradius: R = 21.87553937352

Vertex coordinates: A[40.17768308898; 0] B[0; 0] C[-0.59326383266; 15.78988815251]
Centroid: CG[13.19547308544; 5.26329605084]
Coordinates of the circumscribed circle: U[20.08884154449; 8.66595852085]
Coordinates of the inscribed circle: I[6.12884154449; 6.36327621592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.83° = 158°49'48″ = 0.36994862026 rad
∠ B' = β' = 87.85504024042° = 87°51'1″ = 1.60883138824 rad
∠ C' = γ' = 113.3219597596° = 113°19'11″ = 1.16437925685 rad




How did we calculate this triangle?

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

a = 15.8 ; ; b = 43.72 ; ; alpha = 21.17° ; ;

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 ; ; ; ; 15.8**2 = 43.72**2 + c**2 - 2 * 43.72 * c * cos 21° 10'12" ; ; ; ; ; ; c**2 -81.539c +1661.798 =0 ; ; a=1; b=-81.539; c=1661.798 ; ; D = b**2 - 4ac = 81.539**2 - 4 * 1 * 1661.798 = 1.40488074458 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 81.54 ± sqrt{ 1.4 } }{ 2 } ; ; c_{1,2} = 40.76946922 ± 0.592638326591 ; ; c_{1} = 41.3621075466 ; ; c_{2} = 40.1768308934 ; ;
 ; ; text{ Factored form: } ; ; (c -41.3621075466) (c -40.1768308934) = 0 ; ; ; ; c > 0 ; ; ; ; c = 41.362 ; ;


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 = 15.8 ; ; b = 43.72 ; ; c = 40.18 ; ;

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

p = a+b+c = 15.8+43.72+40.18 = 99.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 99.7 }{ 2 } = 49.85 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 49.85 * (49.85-15.8)(49.85-43.72)(49.85-40.18) } ; ; T = sqrt{ 100599.1 } = 317.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 317.17 }{ 15.8 } = 40.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 317.17 }{ 43.72 } = 14.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 317.17 }{ 40.18 } = 15.79 ; ;

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{ 43.72**2+40.18**2-15.8**2 }{ 2 * 43.72 * 40.18 } ) = 21° 10'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.8**2+40.18**2-43.72**2 }{ 2 * 15.8 * 40.18 } ) = 92° 8'59" ; ; gamma = 180° - alpha - beta = 180° - 21° 10'12" - 92° 8'59" = 66° 40'49" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 317.17 }{ 49.85 } = 6.36 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.8 }{ 2 * sin 21° 10'12" } = 21.88 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.72**2+2 * 40.18**2 - 15.8**2 } }{ 2 } = 41.236 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40.18**2+2 * 15.8**2 - 43.72**2 } }{ 2 } = 21.308 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.72**2+2 * 15.8**2 - 40.18**2 } }{ 2 } = 26.019 ; ;







#2 Acute scalene triangle.

Sides: a = 15.8   b = 43.72   c = 41.3622107543

Area: T = 326.5310707812
Perimeter: p = 100.8822107543
Semiperimeter: s = 50.44110537715

Angle ∠ A = α = 21.17° = 21°10'12″ = 0.36994862026 rad
Angle ∠ B = β = 87.85504024042° = 87°51'1″ = 1.53332787712 rad
Angle ∠ C = γ = 70.98795975958° = 70°58'47″ = 1.23988276798 rad

Height: ha = 41.33330009889
Height: hb = 14.93773608331
Height: hc = 15.78988815251

Median: ma = 41.81877135936
Median: mb = 22.41436648096
Median: mc = 25.55106010673

Inradius: r = 6.47435108289
Circumradius: R = 21.87553937352

Vertex coordinates: A[41.3622107543; 0] B[0; 0] C[0.59326383266; 15.78988815251]
Centroid: CG[13.98549152899; 5.26329605084]
Coordinates of the circumscribed circle: U[20.68110537715; 7.12992963166]
Coordinates of the inscribed circle: I[6.72110537715; 6.47435108289]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.83° = 158°49'48″ = 0.36994862026 rad
∠ B' = β' = 92.15495975958° = 92°8'59″ = 1.53332787712 rad
∠ C' = γ' = 109.0220402404° = 109°1'13″ = 1.23988276798 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 15.8 ; ; b = 43.72 ; ; alpha = 21.17° ; ; : 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 ; ; ; ; 15.8**2 = 43.72**2 + c**2 - 2 * 43.72 * c * cos 21° 10'12" ; ; ; ; ; ; c**2 -81.539c +1661.798 =0 ; ; a=1; b=-81.539; c=1661.798 ; ; D = b**2 - 4ac = 81.539**2 - 4 * 1 * 1661.798 = 1.40488074458 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 81.54 ± sqrt{ 1.4 } }{ 2 } ; ; c_{1,2} = 40.76946922 ± 0.592638326591 ; ; c_{1} = 41.3621075466 ; ; c_{2} = 40.1768308934 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -41.3621075466) (c -40.1768308934) = 0 ; ; ; ; c > 0 ; ; ; ; c = 41.362 ; ; : 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 = 15.8 ; ; b = 43.72 ; ; c = 41.36 ; ;

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

p = a+b+c = 15.8+43.72+41.36 = 100.88 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 100.88 }{ 2 } = 50.44 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 50.44 * (50.44-15.8)(50.44-43.72)(50.44-41.36) } ; ; T = sqrt{ 106622.3 } = 326.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 326.53 }{ 15.8 } = 41.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 326.53 }{ 43.72 } = 14.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 326.53 }{ 41.36 } = 15.79 ; ;

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{ 43.72**2+41.36**2-15.8**2 }{ 2 * 43.72 * 41.36 } ) = 21° 10'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.8**2+41.36**2-43.72**2 }{ 2 * 15.8 * 41.36 } ) = 87° 51'1" ; ; gamma = 180° - alpha - beta = 180° - 21° 10'12" - 87° 51'1" = 70° 58'47" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 326.53 }{ 50.44 } = 6.47 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.8 }{ 2 * sin 21° 10'12" } = 21.88 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.72**2+2 * 41.36**2 - 15.8**2 } }{ 2 } = 41.818 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 41.36**2+2 * 15.8**2 - 43.72**2 } }{ 2 } = 22.414 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.72**2+2 * 15.8**2 - 41.36**2 } }{ 2 } = 25.551 ; ;
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