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=181.4; b=248.6; c=103.5844320765 and a=181.4; b=248.6; c=278.9611138004.

#1 Obtuse scalene triangle.

Sides: a = 181.4   b = 248.6   c = 103.5844320765

Area: T = 8224.475488163
Perimeter: p = 533.5844320765
Semiperimeter: s = 266.7922160383

Angle ∠ A = α = 39.7° = 39°42' = 0.6932895713 rad
Angle ∠ B = β = 118.9087571067° = 118°54'27″ = 2.07553286207 rad
Angle ∠ C = γ = 21.39224289334° = 21°23'33″ = 0.37333683199 rad

Height: ha = 90.67877825979
Height: hb = 66.16663305039
Height: hc = 158.7987679434

Median: ma = 167.4549531962
Median: mb = 79.79656499702
Median: mc = 211.3566410177

Inradius: r = 30.8277273447
Circumradius: R = 141.9922125328

Vertex coordinates: A[103.5844320765; 0] B[0; 0] C[-87.68884086197; 158.7987679434]
Centroid: CG[5.29986373817; 52.93325598115]
Coordinates of the circumscribed circle: U[51.79221603825; 132.2099439065]
Coordinates of the inscribed circle: I[18.19221603825; 30.8277273447]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.3° = 140°18' = 0.6932895713 rad
∠ B' = β' = 61.09224289334° = 61°5'33″ = 2.07553286207 rad
∠ C' = γ' = 158.6087571067° = 158°36'27″ = 0.37333683199 rad




How did we calculate this triangle?

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

a = 181.4 ; ; b = 248.6 ; ; alpha = 39.7° ; ;

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 ; ; ; ; 181.4**2 = 248.6**2 + c**2 - 2 * 248.6 * c * cos 39° 42' ; ; ; ; ; ; c**2 -382.545c +28896 =0 ; ; p=1; q=-382.545; r=28896 ; ; D = q**2 - 4pr = 382.545**2 - 4 * 1 * 28896 = 30757.028025 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 382.55 ± sqrt{ 30757.03 } }{ 2 } ; ; c_{1,2} = 191.27272938 ± 87.6884086197 ; ; c_{1} = 278.961138 ; ; c_{2} = 103.58432076 ; ;
 ; ; text{ Factored form: } ; ; (c -278.961138) (c -103.58432076) = 0 ; ; ; ; c > 0 ; ; ; ; c = 278.961 ; ;
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 = 181.4 ; ; b = 248.6 ; ; c = 103.58 ; ;

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

p = a+b+c = 181.4+248.6+103.58 = 533.58 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 533.58 }{ 2 } = 266.79 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 266.79 * (266.79-181.4)(266.79-248.6)(266.79-103.58) } ; ; T = sqrt{ 67641987.08 } = 8224.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8224.47 }{ 181.4 } = 90.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8224.47 }{ 248.6 } = 66.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8224.47 }{ 103.58 } = 158.8 ; ;

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{ 248.6**2+103.58**2-181.4**2 }{ 2 * 248.6 * 103.58 } ) = 39° 42' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 181.4**2+103.58**2-248.6**2 }{ 2 * 181.4 * 103.58 } ) = 118° 54'27" ; ; gamma = 180° - alpha - beta = 180° - 39° 42' - 118° 54'27" = 21° 23'33" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8224.47 }{ 266.79 } = 30.83 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 181.4 }{ 2 * sin 39° 42' } = 141.99 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 248.6**2+2 * 103.58**2 - 181.4**2 } }{ 2 } = 167.45 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 103.58**2+2 * 181.4**2 - 248.6**2 } }{ 2 } = 79.796 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 248.6**2+2 * 181.4**2 - 103.58**2 } }{ 2 } = 211.356 ; ;







#2 Acute scalene triangle.

Sides: a = 181.4   b = 248.6   c = 278.9611138004

Area: T = 22149.19106837
Perimeter: p = 708.9611138004
Semiperimeter: s = 354.4810569002

Angle ∠ A = α = 39.7° = 39°42' = 0.6932895713 rad
Angle ∠ B = β = 61.09224289334° = 61°5'33″ = 1.06662640329 rad
Angle ∠ C = γ = 79.20875710666° = 79°12'27″ = 1.38224329076 rad

Height: ha = 244.2032763878
Height: hb = 178.1911397295
Height: hc = 158.7987679434

Median: ma = 248.1621536622
Median: mb = 199.7880249921
Median: mc = 167.0330329195

Inradius: r = 62.48435114265
Circumradius: R = 141.9922125328

Vertex coordinates: A[278.9611138004; 0] B[0; 0] C[87.68884086197; 158.7987679434]
Centroid: CG[122.2176515541; 52.93325598115]
Coordinates of the circumscribed circle: U[139.4810569002; 26.58882403689]
Coordinates of the inscribed circle: I[105.8810569002; 62.48435114265]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.3° = 140°18' = 0.6932895713 rad
∠ B' = β' = 118.9087571067° = 118°54'27″ = 1.06662640329 rad
∠ C' = γ' = 100.7922428933° = 100°47'33″ = 1.38224329076 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 181.4 ; ; b = 248.6 ; ; alpha = 39.7° ; ; : 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 ; ; ; ; 181.4**2 = 248.6**2 + c**2 - 2 * 248.6 * c * cos 39° 42' ; ; ; ; ; ; c**2 -382.545c +28896 =0 ; ; p=1; q=-382.545; r=28896 ; ; D = q**2 - 4pr = 382.545**2 - 4 * 1 * 28896 = 30757.028025 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 382.55 ± sqrt{ 30757.03 } }{ 2 } ; ; c_{1,2} = 191.27272938 ± 87.6884086197 ; ; c_{1} = 278.961138 ; ; c_{2} = 103.58432076 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -278.961138) (c -103.58432076) = 0 ; ; ; ; c > 0 ; ; ; ; c = 278.961 ; ; : 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 = 181.4 ; ; b = 248.6 ; ; c = 278.96 ; ;

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

p = a+b+c = 181.4+248.6+278.96 = 708.96 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 708.96 }{ 2 } = 354.48 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 354.48 * (354.48-181.4)(354.48-248.6)(354.48-278.96) } ; ; T = sqrt{ 490586647.94 } = 22149.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22149.19 }{ 181.4 } = 244.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22149.19 }{ 248.6 } = 178.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22149.19 }{ 278.96 } = 158.8 ; ;

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{ 248.6**2+278.96**2-181.4**2 }{ 2 * 248.6 * 278.96 } ) = 39° 42' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 181.4**2+278.96**2-248.6**2 }{ 2 * 181.4 * 278.96 } ) = 61° 5'33" ; ; gamma = 180° - alpha - beta = 180° - 39° 42' - 61° 5'33" = 79° 12'27" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22149.19 }{ 354.48 } = 62.48 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 181.4 }{ 2 * sin 39° 42' } = 141.99 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 248.6**2+2 * 278.96**2 - 181.4**2 } }{ 2 } = 248.162 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 278.96**2+2 * 181.4**2 - 248.6**2 } }{ 2 } = 199.78 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 248.6**2+2 * 181.4**2 - 278.96**2 } }{ 2 } = 167.03 ; ;
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