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=77.73303539385; b=123; c=165 and a=155.6154883853; b=123; c=165.

#1 Obtuse scalene triangle.

Sides: a = 77.73303539385   b = 123   c = 165

Area: T = 4534.502198085
Perimeter: p = 365.7330353939
Semiperimeter: s = 182.8655176969

Angle ∠ A = α = 26.54223402826° = 26°32'32″ = 0.46332512291 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 108.4587659717° = 108°27'28″ = 1.89329432611 rad

Height: ha = 116.6732618896
Height: hb = 73.73217395261
Height: hc = 54.9643660374

Median: ma = 140.2377291828
Median: mb = 113.3633371341
Median: mc = 61.47656371395

Inradius: r = 24.79769682145
Circumradius: R = 86.97441340859

Vertex coordinates: A[165; 0] B[0; 0] C[54.9643660374; 54.9643660374]
Centroid: CG[73.32112201247; 18.32112201247]
Coordinates of the circumscribed circle: U[82.5; -27.5366339626]
Coordinates of the inscribed circle: I[59.86551769693; 24.79769682145]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4587659717° = 153°27'28″ = 0.46332512291 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 71.54223402826° = 71°32'32″ = 1.89329432611 rad




How did we calculate this triangle?

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

b = 123 ; ; c = 165 ; ; beta = 45° ; ;

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 ; ; ; ; 123**2 = 165**2 + a**2 - 2 * 165 * a * cos 45° ; ; ; ; ; ; a**2 -233.345a +12096 =0 ; ; p=1; q=-233.345; r=12096 ; ; D = q**2 - 4pr = 233.345**2 - 4 * 1 * 12096 = 6066 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 233.35 ± sqrt{ 6066 } }{ 2 } ; ; a_{1,2} = 116.6726189 ± 38.9422649572 ; ; a_{1} = 155.614883857 ; ; a_{2} = 77.7303539428 ; ; ; ; text{ Factored form: } ; ; (a -155.614883857) (a -77.7303539428) = 0 ; ; ; ; a > 0 ; ; ; ; a = 155.615 ; ;


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 = 77.73 ; ; b = 123 ; ; c = 165 ; ;

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

p = a+b+c = 77.73+123+165 = 365.73 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 365.73 }{ 2 } = 182.87 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 182.87 * (182.87-77.73)(182.87-123)(182.87-165) } ; ; T = sqrt{ 20561708.21 } = 4534.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4534.5 }{ 77.73 } = 116.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4534.5 }{ 123 } = 73.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4534.5 }{ 165 } = 54.96 ; ;

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{ 123**2+165**2-77.73**2 }{ 2 * 123 * 165 } ) = 26° 32'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 77.73**2+165**2-123**2 }{ 2 * 77.73 * 165 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 26° 32'32" - 45° = 108° 27'28" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4534.5 }{ 182.87 } = 24.8 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 77.73 }{ 2 * sin 26° 32'32" } = 86.97 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 123**2+2 * 165**2 - 77.73**2 } }{ 2 } = 140.237 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 165**2+2 * 77.73**2 - 123**2 } }{ 2 } = 113.363 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 123**2+2 * 77.73**2 - 165**2 } }{ 2 } = 61.476 ; ;







#2 Acute scalene triangle.

Sides: a = 155.6154883853   b = 123   c = 165

Area: T = 9077.998801915
Perimeter: p = 443.6154883853
Semiperimeter: s = 221.8077441926

Angle ∠ A = α = 63.45876597174° = 63°27'28″ = 1.10875450977 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 71.54223402826° = 71°32'32″ = 1.24986493925 rad

Height: ha = 116.6732618896
Height: hb = 147.6109723889
Height: hc = 110.0366339626

Median: ma = 122.9765615391
Median: mb = 148.1165650889
Median: mc = 113.4299476056

Inradius: r = 40.92773825094
Circumradius: R = 86.97441340859

Vertex coordinates: A[165; 0] B[0; 0] C[110.0366339626; 110.0366339626]
Centroid: CG[91.67987798753; 36.67987798753]
Coordinates of the circumscribed circle: U[82.5; 27.5366339626]
Coordinates of the inscribed circle: I[98.80774419265; 40.92773825094]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5422340283° = 116°32'32″ = 1.10875450977 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 108.4587659717° = 108°27'28″ = 1.24986493925 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 123 ; ; c = 165 ; ; beta = 45° ; ; : 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 ; ; ; ; 123**2 = 165**2 + a**2 - 2 * 165 * a * cos 45° ; ; ; ; ; ; a**2 -233.345a +12096 =0 ; ; p=1; q=-233.345; r=12096 ; ; D = q**2 - 4pr = 233.345**2 - 4 * 1 * 12096 = 6066 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 233.35 ± sqrt{ 6066 } }{ 2 } ; ; a_{1,2} = 116.6726189 ± 38.9422649572 ; ; a_{1} = 155.614883857 ; ; a_{2} = 77.7303539428 ; ; ; ; text{ Factored form: } ; ; (a -155.614883857) (a -77.7303539428) = 0 ; ; ; ; a > 0 ; ; ; ; a = 155.615 ; ; : 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 = 155.61 ; ; b = 123 ; ; c = 165 ; ;

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

p = a+b+c = 155.61+123+165 = 443.61 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 443.61 }{ 2 } = 221.81 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 221.81 * (221.81-155.61)(221.81-123)(221.81-165) } ; ; T = sqrt{ 82410048.04 } = 9078 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9078 }{ 155.61 } = 116.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9078 }{ 123 } = 147.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9078 }{ 165 } = 110.04 ; ;

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{ 123**2+165**2-155.61**2 }{ 2 * 123 * 165 } ) = 63° 27'28" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 155.61**2+165**2-123**2 }{ 2 * 155.61 * 165 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 63° 27'28" - 45° = 71° 32'32" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9078 }{ 221.81 } = 40.93 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 155.61 }{ 2 * sin 63° 27'28" } = 86.97 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 123**2+2 * 165**2 - 155.61**2 } }{ 2 } = 122.976 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 165**2+2 * 155.61**2 - 123**2 } }{ 2 } = 148.116 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 123**2+2 * 155.61**2 - 165**2 } }{ 2 } = 113.429 ; ;
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