Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=24.15; b=12.5; c=12.50220682648 and a=24.15; b=12.5; c=34.1522149145.

#1 Obtuse scalene triangle.

Sides: a = 24.15   b = 12.5   c = 12.50220682648

Area: T = 39.0721963444
Perimeter: p = 49.15220682648
Semiperimeter: s = 24.57660341324

Angle ∠ A = α = 149.9977459766° = 149°59'51″ = 2.61879495425 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 15.00325402344° = 15°9″ = 0.26218437233 rad

Height: ha = 3.23657733701
Height: hb = 6.2521514151
Height: hc = 6.25504799392

Median: ma = 3.23657735472
Median: mb = 18.18551479359
Median: mc = 18.18440815626

Inradius: r = 1.59898400545
Circumradius: R = 24.14881456572

Vertex coordinates: A[12.50220682648; 0] B[0; 0] C[23.32771087049; 6.25504799392]
Centroid: CG[11.94330589899; 2.08334933131]
Coordinates of the circumscribed circle: U[6.25110341324; 23.32550404278]
Coordinates of the inscribed circle: I[12.07660341324; 1.59898400545]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00325402344° = 30°9″ = 2.61879495425 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 164.9977459766° = 164°59'51″ = 0.26218437233 rad


How did we calculate this triangle?

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

a = 24.15 ; ; b = 12.5 ; ; beta = 15° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 12.5**2 = 24.15**2 + c**2 - 2 * 24.15 * c * cos 15° ; ; ; ; ; ; c**2 -46.654c +426.973 =0 ; ; p=1; q=-46.654; r=426.973 ; ; D = q**2 - 4pr = 46.654**2 - 4 * 1 * 426.973 = 468.726002117 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 46.65 ± sqrt{ 468.73 } }{ 2 } ; ; c_{1,2} = 23.3271087 ± 10.8250404401 ; ; c_{1} = 34.1521491401 ; ; c_{2} = 12.5020682599 ; ;
 ; ; text{ Factored form: } ; ; (c -34.1521491401) (c -12.5020682599) = 0 ; ; ; ; c > 0 ; ; ; ; c = 34.152 ; ;
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 = 24.15 ; ; b = 12.5 ; ; c = 12.5 ; ;

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

p = a+b+c = 24.15+12.5+12.5 = 49.15 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49.15 }{ 2 } = 24.58 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.58 * (24.58-24.15)(24.58-12.5)(24.58-12.5) } ; ; T = sqrt{ 1526.62 } = 39.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 39.07 }{ 24.15 } = 3.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 39.07 }{ 12.5 } = 6.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 39.07 }{ 12.5 } = 6.25 ; ;

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{ 12.5**2+12.5**2-24.15**2 }{ 2 * 12.5 * 12.5 } ) = 149° 59'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.15**2+12.5**2-12.5**2 }{ 2 * 24.15 * 12.5 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 149° 59'51" - 15° = 15° 9" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 39.07 }{ 24.58 } = 1.59 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.15 }{ 2 * sin 149° 59'51" } = 24.15 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 12.5**2 - 24.15**2 } }{ 2 } = 3.236 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 24.15**2 - 12.5**2 } }{ 2 } = 18.185 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 24.15**2 - 12.5**2 } }{ 2 } = 18.184 ; ;





#2 Obtuse scalene triangle.

Sides: a = 24.15   b = 12.5   c = 34.1522149145

Area: T = 106.7343661556
Perimeter: p = 70.8022149145
Semiperimeter: s = 35.40110745725

Angle ∠ A = α = 30.00325402344° = 30°9″ = 0.52436431111 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 134.9977459766° = 134°59'51″ = 2.35661501547 rad

Height: ha = 8.83992266299
Height: hb = 17.0777385849
Height: hc = 6.25504799392

Median: ma = 22.7054713621
Median: mb = 28.90990538692
Median: mc = 8.84399053838

Inradius: r = 3.01549836649
Circumradius: R = 24.14881456572

Vertex coordinates: A[34.1522149145; 0] B[0; 0] C[23.32771087049; 6.25504799392]
Centroid: CG[19.16597526166; 2.08334933131]
Coordinates of the circumscribed circle: U[17.07660745725; -17.07545604886]
Coordinates of the inscribed circle: I[22.90110745725; 3.01549836649]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9977459766° = 149°59'51″ = 0.52436431111 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 45.00325402344° = 45°9″ = 2.35661501547 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 24.15 ; ; b = 12.5 ; ; beta = 15° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 12.5**2 = 24.15**2 + c**2 - 2 * 24.15 * c * cos 15° ; ; ; ; ; ; c**2 -46.654c +426.973 =0 ; ; p=1; q=-46.654; r=426.973 ; ; D = q**2 - 4pr = 46.654**2 - 4 * 1 * 426.973 = 468.726002117 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 46.65 ± sqrt{ 468.73 } }{ 2 } ; ; c_{1,2} = 23.3271087 ± 10.8250404401 ; ; c_{1} = 34.1521491401 ; ; c_{2} = 12.5020682599 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -34.1521491401) (c -12.5020682599) = 0 ; ; ; ; c > 0 ; ; ; ; c = 34.152 ; ; : 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 = 24.15 ; ; b = 12.5 ; ; c = 34.15 ; ;

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

p = a+b+c = 24.15+12.5+34.15 = 70.8 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70.8 }{ 2 } = 35.4 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.4 * (35.4-24.15)(35.4-12.5)(35.4-34.15) } ; ; T = sqrt{ 11392.07 } = 106.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.73 }{ 24.15 } = 8.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.73 }{ 12.5 } = 17.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.73 }{ 34.15 } = 6.25 ; ;

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{ 12.5**2+34.15**2-24.15**2 }{ 2 * 12.5 * 34.15 } ) = 30° 9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.15**2+34.15**2-12.5**2 }{ 2 * 24.15 * 34.15 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 30° 9" - 15° = 134° 59'51" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.73 }{ 35.4 } = 3.01 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.15 }{ 2 * sin 30° 9" } = 24.15 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 34.15**2 - 24.15**2 } }{ 2 } = 22.705 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.15**2+2 * 24.15**2 - 12.5**2 } }{ 2 } = 28.909 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 24.15**2 - 34.15**2 } }{ 2 } = 8.84 ; ;
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