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=113.194354285; b=265; c=172 and a=359.043991497; b=265; c=172.

#1 Obtuse scalene triangle.

Sides: a = 113.194354285   b = 265   c = 172

Area: T = 6809.015508386
Perimeter: p = 550.194354285
Semiperimeter: s = 275.0976771425

Angle ∠ A = α = 17.38438646978° = 17°23'2″ = 0.30334056757 rad
Angle ∠ B = β = 135.6166135302° = 135°36'58″ = 2.36769480799 rad
Angle ∠ C = γ = 27° = 0.4711238898 rad

Height: ha = 120.3077482431
Height: hb = 51.38987930858
Height: hc = 79.17545939984

Median: ma = 216.1054848313
Median: mb = 60.35501372945
Median: mc = 184.7243818365

Inradius: r = 24.75113449489
Circumradius: R = 189.4311276754

Vertex coordinates: A[172; 0] B[0; 0] C[-80.89659937705; 79.17545939984]
Centroid: CG[30.36880020765; 26.39215313328]
Coordinates of the circumscribed circle: U[86; 168.7854503473]
Coordinates of the inscribed circle: I[10.0976771425; 24.75113449489]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.6166135302° = 162°36'58″ = 0.30334056757 rad
∠ B' = β' = 44.38438646978° = 44°23'2″ = 2.36769480799 rad
∠ C' = γ' = 153° = 0.4711238898 rad


How did we calculate this triangle?

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

b = 265 ; ; c = 172 ; ; gamma = 27° ; ;

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 ; ; ; ; 172**2 = 265**2 + a**2 - 2 * 265 * a * cos 27° ; ; ; ; ; ; a**2 -472.233a +40641 =0 ; ; p=1; q=-472.233; r=40641 ; ; D = q**2 - 4pr = 472.233**2 - 4 * 1 * 40641 = 60440.4386845 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 472.23 ± sqrt{ 60440.44 } }{ 2 } ; ; a_{1,2} = 236.11672891 ± 122.92318606 ; ; a_{1} = 359.03991497 ; ; a_{2} = 113.19354285 ; ;
 ; ; text{ Factored form: } ; ; (a -359.03991497) (a -113.19354285) = 0 ; ; ; ; a > 0 ; ; ; ; a = 359.04 ; ;
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 = 113.19 ; ; b = 265 ; ; c = 172 ; ;

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

p = a+b+c = 113.19+265+172 = 550.19 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 550.19 }{ 2 } = 275.1 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 275.1 * (275.1-113.19)(275.1-265)(275.1-172) } ; ; T = sqrt{ 46362686.41 } = 6809.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6809.02 }{ 113.19 } = 120.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6809.02 }{ 265 } = 51.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6809.02 }{ 172 } = 79.17 ; ;

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{ 265**2+172**2-113.19**2 }{ 2 * 265 * 172 } ) = 17° 23'2" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 113.19**2+172**2-265**2 }{ 2 * 113.19 * 172 } ) = 135° 36'58" ; ; gamma = 180° - alpha - beta = 180° - 17° 23'2" - 135° 36'58" = 27° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6809.02 }{ 275.1 } = 24.75 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 113.19 }{ 2 * sin 17° 23'2" } = 189.43 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 265**2+2 * 172**2 - 113.19**2 } }{ 2 } = 216.105 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 172**2+2 * 113.19**2 - 265**2 } }{ 2 } = 60.35 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 265**2+2 * 113.19**2 - 172**2 } }{ 2 } = 184.724 ; ;





#2 Obtuse scalene triangle.

Sides: a = 359.043991497   b = 265   c = 172

Area: T = 21597.59441311
Perimeter: p = 796.043991497
Semiperimeter: s = 398.0219957485

Angle ∠ A = α = 108.6166135302° = 108°36'58″ = 1.89657091818 rad
Angle ∠ B = β = 44.38438646978° = 44°23'2″ = 0.77546445737 rad
Angle ∠ C = γ = 27° = 0.4711238898 rad

Height: ha = 120.3077482431
Height: hb = 163.0010710424
Height: hc = 251.1354815478

Median: ma = 132.9555198712
Median: mb = 248.3765885043
Median: mc = 303.5977315981

Inradius: r = 54.26325909203
Circumradius: R = 189.4311276754

Vertex coordinates: A[172; 0] B[0; 0] C[256.5954943435; 251.1354815478]
Centroid: CG[142.8654981145; 83.71216051594]
Coordinates of the circumscribed circle: U[86; 168.7854503473]
Coordinates of the inscribed circle: I[133.0219957485; 54.26325909203]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 71.38438646978° = 71°23'2″ = 1.89657091818 rad
∠ B' = β' = 135.6166135302° = 135°36'58″ = 0.77546445737 rad
∠ C' = γ' = 153° = 0.4711238898 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 265 ; ; c = 172 ; ; gamma = 27° ; ; : 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 ; ; ; ; 172**2 = 265**2 + a**2 - 2 * 265 * a * cos 27° ; ; ; ; ; ; a**2 -472.233a +40641 =0 ; ; p=1; q=-472.233; r=40641 ; ; D = q**2 - 4pr = 472.233**2 - 4 * 1 * 40641 = 60440.4386845 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 472.23 ± sqrt{ 60440.44 } }{ 2 } ; ; a_{1,2} = 236.11672891 ± 122.92318606 ; ; a_{1} = 359.03991497 ; ; a_{2} = 113.19354285 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (a -359.03991497) (a -113.19354285) = 0 ; ; ; ; a > 0 ; ; ; ; a = 359.04 ; ; : 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 = 359.04 ; ; b = 265 ; ; c = 172 ; ;

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

p = a+b+c = 359.04+265+172 = 796.04 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 796.04 }{ 2 } = 398.02 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 398.02 * (398.02-359.04)(398.02-265)(398.02-172) } ; ; T = sqrt{ 466456072.25 } = 21597.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21597.59 }{ 359.04 } = 120.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21597.59 }{ 265 } = 163 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21597.59 }{ 172 } = 251.13 ; ;

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{ 265**2+172**2-359.04**2 }{ 2 * 265 * 172 } ) = 108° 36'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 359.04**2+172**2-265**2 }{ 2 * 359.04 * 172 } ) = 44° 23'2" ; ; gamma = 180° - alpha - beta = 180° - 108° 36'58" - 44° 23'2" = 27° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21597.59 }{ 398.02 } = 54.26 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 359.04 }{ 2 * sin 108° 36'58" } = 189.43 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 265**2+2 * 172**2 - 359.04**2 } }{ 2 } = 132.955 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 172**2+2 * 359.04**2 - 265**2 } }{ 2 } = 248.376 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 265**2+2 * 359.04**2 - 172**2 } }{ 2 } = 303.597 ; ;
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