Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=45; b=70.35218356535; c=99 and a=45; b=110.533016496; c=99.

#1 Obtuse scalene triangle.

Sides: a = 45   b = 70.35218356535   c = 99

Area: T = 1416.426613866
Perimeter: p = 214.3521835653
Semiperimeter: s = 107.1765917827

Angle ∠ A = α = 24° = 0.41988790205 rad
Angle ∠ B = β = 39.48553976237° = 39°29'7″ = 0.6899150195 rad
Angle ∠ C = γ = 116.5154602376° = 116°30'53″ = 2.03435634381 rad

Height: ha = 62.95222728294
Height: hb = 40.26769276645
Height: hc = 28.61546694679

Median: ma = 82.87990708798
Median: mb = 68.3798759897
Median: mc = 32.20215588117

Inradius: r = 13.21658993119
Circumradius: R = 55.31883500504

Vertex coordinates: A[99; 0] B[0; 0] C[34.7330400102; 28.61546694679]
Centroid: CG[44.5776800034; 9.5388223156]
Coordinates of the circumscribed circle: U[49.5; -24.69655431668]
Coordinates of the inscribed circle: I[36.82440821733; 13.21658993119]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156° = 0.41988790205 rad
∠ B' = β' = 140.5154602376° = 140°30'53″ = 0.6899150195 rad
∠ C' = γ' = 63.48553976237° = 63°29'7″ = 2.03435634381 rad


How did we calculate this triangle?

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

a = 45 ; ; c = 99 ; ; alpha = 24° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 45**2 = 99**2 + b**2 - 2 * 99 * b * cos 24° ; ; ; ; ; ; b**2 -180.882b +7776 =0 ; ; p=1; q=-180.882; r=7776 ; ; D = q**2 - 4pr = 180.882**2 - 4 * 1 * 7776 = 1614.29814585 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 180.88 ± sqrt{ 1614.3 } }{ 2 } ; ; b_{1,2} = 90.44100031 ± 20.0891646532 ; ; b_{1} = 110.530164963 ; ; b_{2} = 70.3518356568 ; ; ; ;
 text{ Factored form: } ; ; (b -110.530164963) (b -70.3518356568) = 0 ; ; ; ; b > 0 ; ; ; ; b = 110.53 ; ;
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 = 45 ; ; b = 70.35 ; ; c = 99 ; ;

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

p = a+b+c = 45+70.35+99 = 214.35 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 214.35 }{ 2 } = 107.18 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 107.18 * (107.18-45)(107.18-70.35)(107.18-99) } ; ; T = sqrt{ 2006263.01 } = 1416.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1416.43 }{ 45 } = 62.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1416.43 }{ 70.35 } = 40.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1416.43 }{ 99 } = 28.61 ; ;

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{ 70.35**2+99**2-45**2 }{ 2 * 70.35 * 99 } ) = 24° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+99**2-70.35**2 }{ 2 * 45 * 99 } ) = 39° 29'7" ; ; gamma = 180° - alpha - beta = 180° - 24° - 39° 29'7" = 116° 30'53" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1416.43 }{ 107.18 } = 13.22 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 24° } = 55.32 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.35**2+2 * 99**2 - 45**2 } }{ 2 } = 82.879 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 99**2+2 * 45**2 - 70.35**2 } }{ 2 } = 68.379 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.35**2+2 * 45**2 - 99**2 } }{ 2 } = 32.202 ; ;






#2 Obtuse scalene triangle.

Sides: a = 45   b = 110.533016496   c = 99

Area: T = 2225.355507859
Perimeter: p = 254.533016496
Semiperimeter: s = 127.265508248

Angle ∠ A = α = 24° = 0.41988790205 rad
Angle ∠ B = β = 92.51546023763° = 92°30'53″ = 1.61546844176 rad
Angle ∠ C = γ = 63.48553976237° = 63°29'7″ = 1.10880292155 rad

Height: ha = 98.90546701596
Height: hb = 40.26769276645
Height: hc = 44.95766682544

Median: ma = 102.4832723827
Median: mb = 53.46774729017
Median: mc = 68.34325832334

Inradius: r = 17.48659830774
Circumradius: R = 55.31883500504

Vertex coordinates: A[99; 0] B[0; 0] C[-1.97443301315; 44.95766682544]
Centroid: CG[32.34218899562; 14.98655560848]
Coordinates of the circumscribed circle: U[49.5; 24.69655431668]
Coordinates of the inscribed circle: I[16.73549175201; 17.48659830774]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156° = 0.41988790205 rad
∠ B' = β' = 87.48553976237° = 87°29'7″ = 1.61546844176 rad
∠ C' = γ' = 116.5154602376° = 116°30'53″ = 1.10880292155 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 45 ; ; c = 99 ; ; alpha = 24° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 45**2 = 99**2 + b**2 - 2 * 99 * b * cos 24° ; ; ; ; ; ; b**2 -180.882b +7776 =0 ; ; p=1; q=-180.882; r=7776 ; ; D = q**2 - 4pr = 180.882**2 - 4 * 1 * 7776 = 1614.29814585 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 180.88 ± sqrt{ 1614.3 } }{ 2 } ; ; b_{1,2} = 90.44100031 ± 20.0891646532 ; ; b_{1} = 110.530164963 ; ; b_{2} = 70.3518356568 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (b -110.530164963) (b -70.3518356568) = 0 ; ; ; ; b > 0 ; ; ; ; b = 110.53 ; ; : 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 = 45 ; ; b = 110.53 ; ; c = 99 ; ;

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

p = a+b+c = 45+110.53+99 = 254.53 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 254.53 }{ 2 } = 127.27 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 127.27 * (127.27-45)(127.27-110.53)(127.27-99) } ; ; T = sqrt{ 4952205.23 } = 2225.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2225.36 }{ 45 } = 98.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2225.36 }{ 110.53 } = 40.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2225.36 }{ 99 } = 44.96 ; ;

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{ 110.53**2+99**2-45**2 }{ 2 * 110.53 * 99 } ) = 24° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+99**2-110.53**2 }{ 2 * 45 * 99 } ) = 92° 30'53" ; ; gamma = 180° - alpha - beta = 180° - 24° - 92° 30'53" = 63° 29'7" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2225.36 }{ 127.27 } = 17.49 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 24° } = 55.32 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 110.53**2+2 * 99**2 - 45**2 } }{ 2 } = 102.483 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 99**2+2 * 45**2 - 110.53**2 } }{ 2 } = 53.467 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 110.53**2+2 * 45**2 - 99**2 } }{ 2 } = 68.343 ; ;
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