Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=12.87702966504; b=49; c=44 and a=36.13297033496; b=49; c=44.

#1 Obtuse scalene triangle.

Sides: a = 12.87702966504   b = 49   c = 44

Area: T = 273.077709441
Perimeter: p = 105.877029665
Semiperimeter: s = 52.93551483252

Angle ∠ A = α = 14.67439575402° = 14°40'26″ = 0.25661088734 rad
Angle ∠ B = β = 105.326604246° = 105°19'34″ = 1.8388286229 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 42.43552447854
Height: hb = 11.14660038535
Height: hc = 12.41325952005

Median: ma = 46.12203736545
Median: mb = 21.22766876345
Median: mc = 28.27222879855

Inradius: r = 5.15987102908
Circumradius: R = 25.40334118443

Vertex coordinates: A[44; 0] B[0; 0] C[-3.40217666379; 12.41325952005]
Centroid: CG[13.5332744454; 4.13875317335]
Coordinates of the circumscribed circle: U[22; 12.70217059222]
Coordinates of the inscribed circle: I[3.93551483252; 5.15987102908]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.326604246° = 165°19'34″ = 0.25661088734 rad
∠ B' = β' = 74.67439575402° = 74°40'26″ = 1.8388286229 rad
∠ C' = γ' = 120° = 1.04771975512 rad




How did we calculate this triangle?

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

b = 49 ; ; c = 44 ; ; gamma = 60° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 44**2 = 49**2 + a**2 - 2 * 49 * a * cos 60° ; ; ; ; ; ; a**2 -49a +465 =0 ; ; p=1; q=-49; r=465 ; ; D = q**2 - 4pr = 49**2 - 4 * 1 * 465 = 541 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 49 ± sqrt{ 541 } }{ 2 } ; ; a_{1,2} = 24.5 ± 11.6297033496 ; ; a_{1} = 36.1297033496 ; ; a_{2} = 12.8702966504 ; ; ; ; text{ Factored form: } ; ; (a -36.1297033496) (a -12.8702966504) = 0 ; ; ; ; a > 0 ; ; ; ; a = 36.13 ; ;


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 = 12.87 ; ; b = 49 ; ; c = 44 ; ;

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

p = a+b+c = 12.87+49+44 = 105.87 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 105.87 }{ 2 } = 52.94 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 52.94 * (52.94-12.87)(52.94-49)(52.94-44) } ; ; T = sqrt{ 74571.1 } = 273.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 273.08 }{ 12.87 } = 42.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 273.08 }{ 49 } = 11.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 273.08 }{ 44 } = 12.41 ; ;

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+44**2-12.87**2 }{ 2 * 49 * 44 } ) = 14° 40'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.87**2+44**2-49**2 }{ 2 * 12.87 * 44 } ) = 105° 19'34" ; ; gamma = 180° - alpha - beta = 180° - 14° 40'26" - 105° 19'34" = 60° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 273.08 }{ 52.94 } = 5.16 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.87 }{ 2 * sin 14° 40'26" } = 25.4 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 44**2 - 12.87**2 } }{ 2 } = 46.12 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 12.87**2 - 49**2 } }{ 2 } = 21.227 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 12.87**2 - 44**2 } }{ 2 } = 28.272 ; ;







#2 Acute scalene triangle.

Sides: a = 36.13297033496   b = 49   c = 44

Area: T = 766.5866402833
Perimeter: p = 129.132970335
Semiperimeter: s = 64.56548516748

Angle ∠ A = α = 45.32660424598° = 45°19'34″ = 0.79110886778 rad
Angle ∠ B = β = 74.67439575402° = 74°40'26″ = 1.30333064246 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 42.43552447854
Height: hb = 31.2899240932
Height: hc = 34.84548364924

Median: ma = 42.92204046342
Median: mb = 31.94441345487
Median: mc = 37.00224017067

Inradius: r = 11.87331226503
Circumradius: R = 25.40334118443

Vertex coordinates: A[44; 0] B[0; 0] C[9.54994939106; 34.84548364924]
Centroid: CG[17.85498313035; 11.61549454975]
Coordinates of the circumscribed circle: U[22; 12.70217059222]
Coordinates of the inscribed circle: I[15.56548516748; 11.87331226503]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.674395754° = 134°40'26″ = 0.79110886778 rad
∠ B' = β' = 105.326604246° = 105°19'34″ = 1.30333064246 rad
∠ C' = γ' = 120° = 1.04771975512 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 49 ; ; c = 44 ; ; gamma = 60° ; ; : Nr. 1

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 44**2 = 49**2 + a**2 - 2 * 49 * a * cos 60° ; ; ; ; ; ; a**2 -49a +465 =0 ; ; p=1; q=-49; r=465 ; ; D = q**2 - 4pr = 49**2 - 4 * 1 * 465 = 541 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 49 ± sqrt{ 541 } }{ 2 } ; ; a_{1,2} = 24.5 ± 11.6297033496 ; ; a_{1} = 36.1297033496 ; ; a_{2} = 12.8702966504 ; ; ; ; text{ Factored form: } ; ; (a -36.1297033496) (a -12.8702966504) = 0 ; ; ; ; a > 0 ; ; ; ; a = 36.13 ; ; : 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 = 36.13 ; ; b = 49 ; ; c = 44 ; ;

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

p = a+b+c = 36.13+49+44 = 129.13 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 129.13 }{ 2 } = 64.56 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.56 * (64.56-36.13)(64.56-49)(64.56-44) } ; ; T = sqrt{ 587654.71 } = 766.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 766.59 }{ 36.13 } = 42.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 766.59 }{ 49 } = 31.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 766.59 }{ 44 } = 34.84 ; ;

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+44**2-36.13**2 }{ 2 * 49 * 44 } ) = 45° 19'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 36.13**2+44**2-49**2 }{ 2 * 36.13 * 44 } ) = 74° 40'26" ; ; gamma = 180° - alpha - beta = 180° - 45° 19'34" - 74° 40'26" = 60° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 766.59 }{ 64.56 } = 11.87 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 36.13 }{ 2 * sin 45° 19'34" } = 25.4 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 44**2 - 36.13**2 } }{ 2 } = 42.92 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 36.13**2 - 49**2 } }{ 2 } = 31.944 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 36.13**2 - 44**2 } }{ 2 } = 37.002 ; ;
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