Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=193.2488396363 and with side c=54.95500032075

#1 Acute scalene triangle.

Sides: a = 162   b = 125   c = 193.2488396363

Area: T = 10061.63216567
Perimeter: p = 480.2488396363
Semiperimeter: s = 240.1244198181

Angle ∠ A = α = 56.41436175293° = 56°24'49″ = 0.98546033688 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 83.58663824707° = 83°35'11″ = 1.45988575839 rad

Height: ha = 124.2187674774
Height: hb = 160.9866106507
Height: hc = 104.1321592769

Median: ma = 141.1522298417
Median: mb = 166.9987668692
Median: mc = 107.6955238176

Inradius: r = 41.90217813818
Circumradius: R = 97.23327391788

Vertex coordinates: A[193.2488396363; 0] B[0; 0] C[124.0999199785; 104.1321592769]
Centroid: CG[105.7832532049; 34.71105309231]
Coordinates of the circumscribed circle: U[96.62441981815; 10.86113946614]
Coordinates of the inscribed circle: I[115.1244198182; 41.90217813818]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.5866382471° = 123°35'11″ = 0.98546033688 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 96.41436175293° = 96°24'49″ = 1.45988575839 rad




How did we calculate this triangle?

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 = 162 ; ; b = 125 ; ; c = 193.25 ; ;

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

p = a+b+c = 162+125+193.25 = 480.25 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 480.25 }{ 2 } = 240.12 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 240.12 * (240.12-162)(240.12-125)(240.12-193.25) } ; ; T = sqrt{ 101236431.59 } = 10061.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10061.63 }{ 162 } = 124.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10061.63 }{ 125 } = 160.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10061.63 }{ 193.25 } = 104.13 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 162**2-125**2-193.25**2 }{ 2 * 125 * 193.25 } ) = 56° 24'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 125**2-162**2-193.25**2 }{ 2 * 162 * 193.25 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 193.25**2-162**2-125**2 }{ 2 * 125 * 162 } ) = 83° 35'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10061.63 }{ 240.12 } = 41.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 162 }{ 2 * sin 56° 24'49" } = 97.23 ; ;





#2 Obtuse scalene triangle.

Sides: a = 162   b = 125   c = 54.95500032075

Area: T = 2861.016567834
Perimeter: p = 341.9550003208
Semiperimeter: s = 170.9755001604

Angle ∠ A = α = 123.5866382471° = 123°35'11″ = 2.15769892847 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 16.41436175293° = 16°24'49″ = 0.28664716681 rad

Height: ha = 35.3211181214
Height: hb = 45.77662508534
Height: hc = 104.1321592769

Median: ma = 52.54876110423
Median: mb = 103.5643996766
Median: mc = 142.0555004441

Inradius: r = 16.73435321041
Circumradius: R = 97.23327391788

Vertex coordinates: A[54.95500032075; 0] B[0; 0] C[124.0999199785; 104.1321592769]
Centroid: CG[59.68330676643; 34.71105309231]
Coordinates of the circumscribed circle: U[27.47550016038; 93.27701981079]
Coordinates of the inscribed circle: I[45.97550016038; 16.73435321041]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.41436175293° = 56°24'49″ = 2.15769892847 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 163.5866382471° = 163°35'11″ = 0.28664716681 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 162 ; ; b = 125 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 125**2 = 162**2 + c**2 -2 * 125 * c * cos (40° ) ; ; ; ; c**2 -248.198c +10619 =0 ; ; p=1; q=-248.198399571; r=10619 ; ; D = q**2 - 4pr = 248.198**2 - 4 * 1 * 10619 = 19126.4455494 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 248.2 ± sqrt{ 19126.45 } }{ 2 } ; ; c_{1,2} = 124.099199785 ± 69.1491965777 ; ;
c_{1} = 193.248396363 ; ; c_{2} = 54.9500032075 ; ; ; ; (c -193.248396363) (c -54.9500032075) = 0 ; ; ; ; c>0 ; ;


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 = 162 ; ; b = 125 ; ; c = 54.95 ; ;

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

p = a+b+c = 162+125+54.95 = 341.95 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 341.95 }{ 2 } = 170.98 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 170.98 * (170.98-162)(170.98-125)(170.98-54.95) } ; ; T = sqrt{ 8185410.71 } = 2861.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2861.02 }{ 162 } = 35.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2861.02 }{ 125 } = 45.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2861.02 }{ 54.95 } = 104.13 ; ;

6. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 162**2-125**2-54.95**2 }{ 2 * 125 * 54.95 } ) = 123° 35'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 125**2-162**2-54.95**2 }{ 2 * 162 * 54.95 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 54.95**2-162**2-125**2 }{ 2 * 125 * 162 } ) = 16° 24'49" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2861.02 }{ 170.98 } = 16.73 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 162 }{ 2 * sin 123° 35'11" } = 97.23 ; ;




Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.