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=54; b=49; c=7.03329551737 and a=54; b=49; c=73.22766859779.

#1 Obtuse scalene triangle.

Sides: a = 54   b = 49   c = 7.03329551737

Area: T = 127.0611070116
Perimeter: p = 110.0332955174
Semiperimeter: s = 55.01664775868

Angle ∠ A = α = 132.4898812327° = 132°29'20″ = 2.31223659972 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 5.51111876725° = 5°30'40″ = 0.09661883706 rad

Height: ha = 4.70659655599
Height: hb = 5.18661661272
Height: hc = 36.13330527434

Median: ma = 22.27662480961
Median: mb = 29.70765856207
Median: mc = 51.44105908343

Inradius: r = 2.31095093632
Circumradius: R = 36.61546754717

Vertex coordinates: A[7.03329551737; 0] B[0; 0] C[40.13298205758; 36.13330527434]
Centroid: CG[15.72109252498; 12.04443509145]
Coordinates of the circumscribed circle: U[3.51664775868; 36.44554228303]
Coordinates of the inscribed circle: I[6.01664775868; 2.31095093632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.51111876725° = 47°30'40″ = 2.31223659972 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 174.4898812327° = 174°29'20″ = 0.09661883706 rad


How did we calculate this triangle?

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

a = 54 ; ; b = 49 ; ; beta = 42° ; ;

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 ; ; ; ; 49**2 = 54**2 + c**2 - 2 * 54 * c * cos 42° ; ; ; ; ; ; c**2 -80.26c +515 =0 ; ; p=1; q=-80.26; r=515 ; ; D = q**2 - 4pr = 80.26**2 - 4 * 1 * 515 = 4381.60999778 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 80.26 ± sqrt{ 4381.61 } }{ 2 } ; ; c_{1,2} = 40.12982058 ± 33.0968654021 ; ; c_{1} = 73.2266859821 ; ; c_{2} = 7.03295517791 ; ; ; ; text{ Factored form: } ; ;
(c -73.2266859821) (c -7.03295517791) = 0 ; ; ; ; c > 0 ; ; ; ; c = 73.227 ; ;
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 = 54 ; ; b = 49 ; ; c = 7.03 ; ;

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

p = a+b+c = 54+49+7.03 = 110.03 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 110.03 }{ 2 } = 55.02 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 55.02 * (55.02-54)(55.02-49)(55.02-7.03) } ; ; T = sqrt{ 16144.52 } = 127.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127.06 }{ 54 } = 4.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127.06 }{ 49 } = 5.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127.06 }{ 7.03 } = 36.13 ; ;

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{ 49**2+7.03**2-54**2 }{ 2 * 49 * 7.03 } ) = 132° 29'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 54**2+7.03**2-49**2 }{ 2 * 54 * 7.03 } ) = 42° ; ; gamma = 180° - alpha - beta = 180° - 132° 29'20" - 42° = 5° 30'40" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127.06 }{ 55.02 } = 2.31 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 54 }{ 2 * sin 132° 29'20" } = 36.61 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 7.03**2 - 54**2 } }{ 2 } = 22.276 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.03**2+2 * 54**2 - 49**2 } }{ 2 } = 29.707 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 54**2 - 7.03**2 } }{ 2 } = 51.441 ; ;





#2 Obtuse scalene triangle.

Sides: a = 54   b = 49   c = 73.22766859779

Area: T = 1322.952185333
Perimeter: p = 176.2276685978
Semiperimeter: s = 88.11333429889

Angle ∠ A = α = 47.51111876725° = 47°30'40″ = 0.82992266564 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 90.48988123275° = 90°29'20″ = 1.57993277113 rad

Height: ha = 48.998821679
Height: hb = 53.99880348298
Height: hc = 36.13330527434

Median: ma = 56.14877850823
Median: mb = 59.48880136637
Median: mc = 36.30437617221

Inradius: r = 15.01442056635
Circumradius: R = 36.61546754717

Vertex coordinates: A[73.22766859779; 0] B[0; 0] C[40.13298205758; 36.13330527434]
Centroid: CG[37.78655021846; 12.04443509145]
Coordinates of the circumscribed circle: U[36.61333429889; -0.31223700869]
Coordinates of the inscribed circle: I[39.11333429889; 15.01442056635]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.4898812327° = 132°29'20″ = 0.82992266564 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 89.51111876725° = 89°30'40″ = 1.57993277113 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 54 ; ; b = 49 ; ; beta = 42° ; ; : 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 ; ; ; ; 49**2 = 54**2 + c**2 - 2 * 54 * c * cos 42° ; ; ; ; ; ; c**2 -80.26c +515 =0 ; ; p=1; q=-80.26; r=515 ; ; D = q**2 - 4pr = 80.26**2 - 4 * 1 * 515 = 4381.60999778 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 80.26 ± sqrt{ 4381.61 } }{ 2 } ; ; c_{1,2} = 40.12982058 ± 33.0968654021 ; ; c_{1} = 73.2266859821 ; ; c_{2} = 7.03295517791 ; ; ; ; text{ Factored form: } ; ; : Nr. 1
(c -73.2266859821) (c -7.03295517791) = 0 ; ; ; ; c > 0 ; ; ; ; c = 73.227 ; ; : 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 = 54 ; ; b = 49 ; ; c = 73.23 ; ;

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

p = a+b+c = 54+49+73.23 = 176.23 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 176.23 }{ 2 } = 88.11 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 88.11 * (88.11-54)(88.11-49)(88.11-73.23) } ; ; T = sqrt{ 1750201.61 } = 1322.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1322.95 }{ 54 } = 49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1322.95 }{ 49 } = 54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1322.95 }{ 73.23 } = 36.13 ; ;

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{ 49**2+73.23**2-54**2 }{ 2 * 49 * 73.23 } ) = 47° 30'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 54**2+73.23**2-49**2 }{ 2 * 54 * 73.23 } ) = 42° ; ; gamma = 180° - alpha - beta = 180° - 47° 30'40" - 42° = 90° 29'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1322.95 }{ 88.11 } = 15.01 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 54 }{ 2 * sin 47° 30'40" } = 36.61 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 73.23**2 - 54**2 } }{ 2 } = 56.148 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 73.23**2+2 * 54**2 - 49**2 } }{ 2 } = 59.488 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 54**2 - 73.23**2 } }{ 2 } = 36.304 ; ;
Calculate another triangle

Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.