Triangle calculator

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

You have entered side a, c and height hc.

Triangle has two solutions: a=405; b=1382.678781137; c=982 and a=405; b=587.2821933953; c=982.

#1 Obtuse scalene triangle.

Sides: a = 405   b = 1382.678781137   c = 982

Area: T = 34370
Perimeter: p = 2769.678781137
Semiperimeter: s = 1384.839890568

Angle ∠ A = α = 2.90219195744° = 2°54'7″ = 0.05106480512 rad
Angle ∠ B = β = 170.0477044832° = 170°2'49″ = 2.96878808156 rad
Angle ∠ C = γ = 7.05110355939° = 7°3'4″ = 0.12330637868 rad

Height: ha = 169.7288395062
Height: hb = 49.71551248359
Height: hc = 70

Median: ma = 1181.971069127
Median: mb = 293.6410966977
Median: mc = 892.6543608644

Inradius: r = 24.81987712368
Circumradius: R = 3999.889938289

Vertex coordinates: A[982; 0] B[0; 0] C[-398.9054750536; 70]
Centroid: CG[194.3655083155; 23.33333333333]
Coordinates of the circumscribed circle: U[491; 3969.63990359]
Coordinates of the inscribed circle: I[2.16110943155; 24.81987712368]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 177.0988080426° = 177°5'53″ = 0.05106480512 rad
∠ B' = β' = 9.95329551683° = 9°57'11″ = 2.96878808156 rad
∠ C' = γ' = 172.9498964406° = 172°56'56″ = 0.12330637868 rad


How did we calculate this triangle?

1. Input data entered: side a, c and height hc.

a = 405 ; ; c = 982 ; ; h_c = 70 ; ;

2. From side c we calculate T:

T = fraction{ c h_c }{ 2 } ; ; ; ; T = fraction{ 982 * 70 }{ 2 } = 34370 ; ;
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 = 405 ; ; b = 1382.68 ; ; c = 982 ; ;

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

p = a+b+c = 405+1382.68+982 = 2769.68 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2769.68 }{ 2 } = 1384.84 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1384.84 * (1384.84-405)(1384.84-1382.68)(1384.84-982) } ; ; T = sqrt{ 1181296900 } = 34370 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34370 }{ 405 } = 169.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34370 }{ 1382.68 } = 49.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34370 }{ 982 } = 70 ; ;

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{ 1382.68**2+982**2-405**2 }{ 2 * 1382.68 * 982 } ) = 2° 54'7" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 405**2+982**2-1382.68**2 }{ 2 * 405 * 982 } ) = 170° 2'49" ; ;
 gamma = 180° - alpha - beta = 180° - 2° 54'7" - 170° 2'49" = 7° 3'4" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34370 }{ 1384.84 } = 24.82 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 405 }{ 2 * sin 2° 54'7" } = 3999.89 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1382.68**2+2 * 982**2 - 405**2 } }{ 2 } = 1181.971 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 982**2+2 * 405**2 - 1382.68**2 } }{ 2 } = 293.641 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1382.68**2+2 * 405**2 - 982**2 } }{ 2 } = 892.654 ; ;



#2 Obtuse scalene triangle.

Sides: a = 405   b = 587.2821933953   c = 982

Area: T = 34370
Perimeter: p = 1974.282193395
Semiperimeter: s = 987.1410966977

Angle ∠ A = α = 6.84655408805° = 6°50'44″ = 0.11994772274 rad
Angle ∠ B = β = 9.95329551683° = 9°57'11″ = 0.1743711838 rad
Angle ∠ C = γ = 163.2021503951° = 163°12'5″ = 2.84884035882 rad

Height: ha = 169.7288395062
Height: hb = 117.0487700646
Height: hc = 70

Median: ma = 783.3329933664
Median: mb = 691.3398905684
Median: mc = 115.6798584768

Inradius: r = 34.81877222401
Circumradius: R = 1698.923273751

Vertex coordinates: A[982; 0] B[0; 0] C[398.9054750536; 70]
Centroid: CG[460.3021583512; 23.33333333333]
Coordinates of the circumscribed circle: U[491; -1626.425475019]
Coordinates of the inscribed circle: I[399.8599033023; 34.81877222401]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173.1544459119° = 173°9'16″ = 0.11994772274 rad
∠ B' = β' = 170.0477044832° = 170°2'49″ = 0.1743711838 rad
∠ C' = γ' = 16.79884960488° = 16°47'55″ = 2.84884035882 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and height hc.

a = 405 ; ; c = 982 ; ; h_c = 70 ; ; : Nr. 1

2. From side c we calculate T:

T = fraction{ c h_c }{ 2 } ; ; ; ; T = fraction{ 982 * 70 }{ 2 } = 34370 ; ; : 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 = 405 ; ; b = 587.28 ; ; c = 982 ; ;

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

p = a+b+c = 405+587.28+982 = 1974.28 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1974.28 }{ 2 } = 987.14 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 987.14 * (987.14-405)(987.14-587.28)(987.14-982) } ; ; T = sqrt{ 1181296900 } = 34370 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34370 }{ 405 } = 169.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34370 }{ 587.28 } = 117.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34370 }{ 982 } = 70 ; ;

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{ 587.28**2+982**2-405**2 }{ 2 * 587.28 * 982 } ) = 6° 50'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 405**2+982**2-587.28**2 }{ 2 * 405 * 982 } ) = 9° 57'11" ; ;
 gamma = 180° - alpha - beta = 180° - 6° 50'44" - 9° 57'11" = 163° 12'5" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34370 }{ 987.14 } = 34.82 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 405 }{ 2 * sin 6° 50'44" } = 1698.92 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 587.28**2+2 * 982**2 - 405**2 } }{ 2 } = 783.33 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 982**2+2 * 405**2 - 587.28**2 } }{ 2 } = 691.339 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 587.28**2+2 * 405**2 - 982**2 } }{ 2 } = 115.679 ; ;
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