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=1.05436078092; b=2.4; c=1.5 and a=3.33114103874; b=2.4; c=1.5.

#1 Obtuse scalene triangle.

Sides: a = 1.05436078092   b = 2.4   c = 1.5

Area: T = 0.51442490841
Perimeter: p = 4.95436078092
Semiperimeter: s = 2.47768039046

Angle ∠ A = α = 16.66003329111° = 16°36'1″ = 0.29897304662 rad
Angle ∠ B = β = 139.4399667089° = 139°23'59″ = 2.43329831669 rad
Angle ∠ C = γ = 24° = 0.41988790205 rad

Height: ha = 0.97661679434
Height: hb = 0.42985409034
Height: hc = 0.68656654455

Median: ma = 1.93106676685
Median: mb = 0.49899435762
Median: mc = 1.6954858315

Inradius: r = 0.20876260794
Circumradius: R = 1.84439450017

Vertex coordinates: A[1.5; 0] B[0; 0] C[-0.87999701948; 0.68656654455]
Centroid: CG[0.23333432684; 0.22985551485]
Coordinates of the circumscribed circle: U[0.75; 1.68545275804]
Coordinates of the inscribed circle: I[0.07768039046; 0.20876260794]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.4399667089° = 163°23'59″ = 0.29897304662 rad
∠ B' = β' = 40.66003329111° = 40°36'1″ = 2.43329831669 rad
∠ C' = γ' = 156° = 0.41988790205 rad




How did we calculate this triangle?

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

b = 2.4 ; ; c = 1.5 ; ; gamma = 24° ; ;

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 ; ; ; ; 1.5**2 = 2.4**2 + a**2 - 2 * 2.4 * a * cos 24° ; ; ; ; ; ; a**2 -4.385a +3.51 =0 ; ; p=1; q=-4.385; r=3.51 ; ; D = q**2 - 4pr = 4.385**2 - 4 * 1 * 3.51 = 5.18838458525 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.39 ± sqrt{ 5.19 } }{ 2 } ; ; a_{1,2} = 2.1925091 ± 1.1389012891 ; ; a_{1} = 3.3314103891 ; ; a_{2} = 1.0536078109 ; ; ; ; text{ Factored form: } ; ; (a -3.3314103891) (a -1.0536078109) = 0 ; ; ; ; a > 0 ; ; ; ; a = 3.331 ; ;


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 = 1.05 ; ; b = 2.4 ; ; c = 1.5 ; ;

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

p = a+b+c = 1.05+2.4+1.5 = 4.95 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.95 }{ 2 } = 2.48 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.48 * (2.48-1.05)(2.48-2.4)(2.48-1.5) } ; ; T = sqrt{ 0.26 } = 0.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.51 }{ 1.05 } = 0.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.51 }{ 2.4 } = 0.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.51 }{ 1.5 } = 0.69 ; ;

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{ 2.4**2+1.5**2-1.05**2 }{ 2 * 2.4 * 1.5 } ) = 16° 36'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.05**2+1.5**2-2.4**2 }{ 2 * 1.05 * 1.5 } ) = 139° 23'59" ; ; gamma = 180° - alpha - beta = 180° - 16° 36'1" - 139° 23'59" = 24° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.51 }{ 2.48 } = 0.21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.05 }{ 2 * sin 16° 36'1" } = 1.84 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 1.5**2 - 1.05**2 } }{ 2 } = 1.931 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.5**2+2 * 1.05**2 - 2.4**2 } }{ 2 } = 0.49 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 1.05**2 - 1.5**2 } }{ 2 } = 1.695 ; ;







#2 Obtuse scalene triangle.

Sides: a = 3.33114103874   b = 2.4   c = 1.5

Area: T = 1.62660080132
Perimeter: p = 7.23114103874
Semiperimeter: s = 3.61657051937

Angle ∠ A = α = 115.4399667089° = 115°23'59″ = 2.01441041464 rad
Angle ∠ B = β = 40.66003329111° = 40°36'1″ = 0.70986094867 rad
Angle ∠ C = γ = 24° = 0.41988790205 rad

Height: ha = 0.97661679434
Height: hb = 1.35550066777
Height: hc = 2.16880106843

Median: ma = 1.10992457832
Median: mb = 2.28878259516
Median: mc = 2.80547544607

Inradius: r = 0.45497070215
Circumradius: R = 1.84439450017

Vertex coordinates: A[1.5; 0] B[0; 0] C[2.52994317232; 2.16880106843]
Centroid: CG[1.34331439077; 0.72326702281]
Coordinates of the circumscribed circle: U[0.75; 1.68545275804]
Coordinates of the inscribed circle: I[1.21657051937; 0.45497070215]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.66003329111° = 64°36'1″ = 2.01441041464 rad
∠ B' = β' = 139.4399667089° = 139°23'59″ = 0.70986094867 rad
∠ C' = γ' = 156° = 0.41988790205 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 2.4 ; ; c = 1.5 ; ; gamma = 24° ; ; : 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 ; ; ; ; 1.5**2 = 2.4**2 + a**2 - 2 * 2.4 * a * cos 24° ; ; ; ; ; ; a**2 -4.385a +3.51 =0 ; ; p=1; q=-4.385; r=3.51 ; ; D = q**2 - 4pr = 4.385**2 - 4 * 1 * 3.51 = 5.18838458525 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.39 ± sqrt{ 5.19 } }{ 2 } ; ; a_{1,2} = 2.1925091 ± 1.1389012891 ; ; a_{1} = 3.3314103891 ; ; a_{2} = 1.0536078109 ; ; ; ; text{ Factored form: } ; ; (a -3.3314103891) (a -1.0536078109) = 0 ; ; ; ; a > 0 ; ; ; ; a = 3.331 ; ; : 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 = 3.33 ; ; b = 2.4 ; ; c = 1.5 ; ;

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

p = a+b+c = 3.33+2.4+1.5 = 7.23 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.23 }{ 2 } = 3.62 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.62 * (3.62-3.33)(3.62-2.4)(3.62-1.5) } ; ; T = sqrt{ 2.64 } = 1.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.63 }{ 3.33 } = 0.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.63 }{ 2.4 } = 1.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.63 }{ 1.5 } = 2.17 ; ;

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{ 2.4**2+1.5**2-3.33**2 }{ 2 * 2.4 * 1.5 } ) = 115° 23'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.33**2+1.5**2-2.4**2 }{ 2 * 3.33 * 1.5 } ) = 40° 36'1" ; ; gamma = 180° - alpha - beta = 180° - 115° 23'59" - 40° 36'1" = 24° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.63 }{ 3.62 } = 0.45 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.33 }{ 2 * sin 115° 23'59" } = 1.84 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 1.5**2 - 3.33**2 } }{ 2 } = 1.109 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.5**2+2 * 3.33**2 - 2.4**2 } }{ 2 } = 2.288 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 3.33**2 - 1.5**2 } }{ 2 } = 2.805 ; ;
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