Triangle calculator

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

You have entered side b, c and area T.

Triangle has two solutions: a=31.11774184616; b=21.9; c=10.4 and a=14.39660504404; b=21.9; c=10.4.

#1 Obtuse scalene triangle.

Sides: a = 31.11774184616   b = 21.9   c = 10.4

Area: T = 62.6
Perimeter: p = 63.41774184616
Semiperimeter: s = 31.70987092308

Angle ∠ A = α = 146.6533467075° = 146°39'12″ = 2.56595858599 rad
Angle ∠ B = β = 22.76600128159° = 22°45'36″ = 0.39772371614 rad
Angle ∠ C = γ = 10.58765201095° = 10°35'11″ = 0.18547696322 rad

Height: ha = 4.0233470011
Height: hb = 5.71768949772
Height: hc = 12.03884615385

Median: ma = 7.19880252202
Median: mb = 20.45329793883
Median: mc = 26.39990883528

Inradius: r = 1.97442210112
Circumradius: R = 28.30439266327

Vertex coordinates: A[10.4; 0] B[0; 0] C[28.69444101787; 12.03884615385]
Centroid: CG[13.03114700596; 4.01328205128]
Coordinates of the circumscribed circle: U[5.2; 27.8222154173]
Coordinates of the inscribed circle: I[9.80987092308; 1.97442210112]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 33.34765329254° = 33°20'48″ = 2.56595858599 rad
∠ B' = β' = 157.2439987184° = 157°14'24″ = 0.39772371614 rad
∠ C' = γ' = 169.413347989° = 169°24'49″ = 0.18547696322 rad


How did we calculate this triangle?

1. Input data entered: side b, c and area T.

b = 21.9 ; ; c = 10.4 ; ; T = 62.6 ; ;

2. From area T, side b and side c we calculate side a - using Heron's formula for the area and solve of the bikvadratic equation:

s = fraction{ a+b+c }{ 2 } ; ; T**2 = s(s-a)(s-b)(s-c) ; ; ; ; s = fraction{ a+21.9+10.4 }{ 2 } = fraction{ a+32.3 }{ 2 } = a/2 + 16.15 ; ; ; ; T**2 = s(s-a)(s-b)(s-c) ; ; T**2 = ( a/2 + 16.15) ( a/2 + 16.15-a) ( a/2 + 16.15-21.9) ( a/2 + 16.15 - 10.4) ; ; ; ; 62.6**2 = ( a/2 + 16.15) ( 16.15-a/2) ( a/2 + (-5.75)) ( a/2 + 5.75) ; ;
62700.16 = ( a + 32.3) ( 32.3-a) ( a + (-11.5)) ( a + 11.5) ; ; ; ; D = b**2 * c**2 - 4 * T**2 = 21.9**2 * 10.4**2 - 4 * 62.6**2 = 36199.578 ; ; ; ; D_1 = -2 * sqrt{ D } + b**2 + c**2 = -2 * sqrt{ 36199.578 } + 21.9**2 + 10.4**2 = 207.246 ; ; D_2 = 2 * sqrt{ D } + b**2 + c**2 = 2 * sqrt{ 36199.578 } + 21.9**2 + 10.4**2 = 968.294 ; ;
 ; ; a_1 = sqrt{ D_1 } = sqrt{ 207.246 } = 14.396 ; ; a_2 = sqrt{ D_2 } = sqrt{ 968.294 } = 31.117 ; ;
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 = 31.12 ; ; b = 21.9 ; ; c = 10.4 ; ;

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

p = a+b+c = 31.12+21.9+10.4 = 63.42 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63.42 }{ 2 } = 31.71 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.71 * (31.71-31.12)(31.71-21.9)(31.71-10.4) } ; ; T = sqrt{ 3918.76 } = 62.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.6 }{ 31.12 } = 4.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.6 }{ 21.9 } = 5.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.6 }{ 10.4 } = 12.04 ; ;

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{ 21.9**2+10.4**2-31.12**2 }{ 2 * 21.9 * 10.4 } ) = 146° 39'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31.12**2+10.4**2-21.9**2 }{ 2 * 31.12 * 10.4 } ) = 22° 45'36" ; ;
 gamma = 180° - alpha - beta = 180° - 146° 39'12" - 22° 45'36" = 10° 35'11" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.6 }{ 31.71 } = 1.97 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 31.12 }{ 2 * sin 146° 39'12" } = 28.3 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.9**2+2 * 10.4**2 - 31.12**2 } }{ 2 } = 7.198 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.4**2+2 * 31.12**2 - 21.9**2 } }{ 2 } = 20.453 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.9**2+2 * 31.12**2 - 10.4**2 } }{ 2 } = 26.399 ; ;



#2 Obtuse scalene triangle.

Sides: a = 14.39660504404   b = 21.9   c = 10.4

Area: T = 62.6
Perimeter: p = 46.69660504404
Semiperimeter: s = 23.34880252202

Angle ∠ A = α = 33.34765329254° = 33°20'48″ = 0.58220067937 rad
Angle ∠ B = β = 123.2555488315° = 123°15'20″ = 2.15112140922 rad
Angle ∠ C = γ = 23.39879787596° = 23°23'53″ = 0.40883717677 rad

Height: ha = 8.69768297672
Height: hb = 5.71768949772
Height: hc = 12.03884615385

Median: ma = 15.55987092308
Median: mb = 6.14882220309
Median: mc = 17.787730261

Inradius: r = 2.68111689387
Circumradius: R = 13.09444267105

Vertex coordinates: A[10.4; 0] B[0; 0] C[-7.89444101787; 12.03884615385]
Centroid: CG[0.83551966071; 4.01328205128]
Coordinates of the circumscribed circle: U[5.2; 12.01876541337]
Coordinates of the inscribed circle: I[1.44880252202; 2.68111689387]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.6533467075° = 146°39'12″ = 0.58220067937 rad
∠ B' = β' = 56.74545116851° = 56°44'40″ = 2.15112140922 rad
∠ C' = γ' = 156.602202124° = 156°36'7″ = 0.40883717677 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and area T.

b = 21.9 ; ; c = 10.4 ; ; T = 62.6 ; ; : Nr. 1

2. From area T, side b and side c we calculate side a - using Heron's formula for the area and solve of the bikvadratic equation:

s = fraction{ a+b+c }{ 2 } ; ; T**2 = s(s-a)(s-b)(s-c) ; ; ; ; s = fraction{ a+21.9+10.4 }{ 2 } = fraction{ a+32.3 }{ 2 } = a/2 + 16.15 ; ; ; ; T**2 = s(s-a)(s-b)(s-c) ; ; T**2 = ( a/2 + 16.15) ( a/2 + 16.15-a) ( a/2 + 16.15-21.9) ( a/2 + 16.15 - 10.4) ; ; ; ; 62.6**2 = ( a/2 + 16.15) ( 16.15-a/2) ( a/2 + (-5.75)) ( a/2 + 5.75) ; ; : Nr. 1
62700.16 = ( a + 32.3) ( 32.3-a) ( a + (-11.5)) ( a + 11.5) ; ; ; ; D = b**2 * c**2 - 4 * T**2 = 21.9**2 * 10.4**2 - 4 * 62.6**2 = 36199.578 ; ; ; ; D_1 = -2 * sqrt{ D } + b**2 + c**2 = -2 * sqrt{ 36199.578 } + 21.9**2 + 10.4**2 = 207.246 ; ; D_2 = 2 * sqrt{ D } + b**2 + c**2 = 2 * sqrt{ 36199.578 } + 21.9**2 + 10.4**2 = 968.294 ; ; : Nr. 1
 ; ; a_1 = sqrt{ D_1 } = sqrt{ 207.246 } = 14.396 ; ; a_2 = sqrt{ D_2 } = sqrt{ 968.294 } = 31.117 ; ; : 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 = 14.4 ; ; b = 21.9 ; ; c = 10.4 ; ;

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

p = a+b+c = 14.4+21.9+10.4 = 46.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46.7 }{ 2 } = 23.35 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.35 * (23.35-14.4)(23.35-21.9)(23.35-10.4) } ; ; T = sqrt{ 3918.76 } = 62.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.6 }{ 14.4 } = 8.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.6 }{ 21.9 } = 5.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.6 }{ 10.4 } = 12.04 ; ;

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{ 21.9**2+10.4**2-14.4**2 }{ 2 * 21.9 * 10.4 } ) = 33° 20'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.4**2+10.4**2-21.9**2 }{ 2 * 14.4 * 10.4 } ) = 123° 15'20" ; ;
 gamma = 180° - alpha - beta = 180° - 33° 20'48" - 123° 15'20" = 23° 23'53" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.6 }{ 23.35 } = 2.68 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14.4 }{ 2 * sin 33° 20'48" } = 13.09 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.9**2+2 * 10.4**2 - 14.4**2 } }{ 2 } = 15.559 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.4**2+2 * 14.4**2 - 21.9**2 } }{ 2 } = 6.148 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.9**2+2 * 14.4**2 - 10.4**2 } }{ 2 } = 17.787 ; ;
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