Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=12.79436569493 and with side c=1.36328706842

#1 Obtuse scalene triangle.

Sides: a = 7.81   b = 6.6   c = 12.79436569493

Area: T = 21.11436831041
Perimeter: p = 27.20436569493
Semiperimeter: s = 13.60218284747

Angle ∠ A = α = 30.00765021269° = 30°23″ = 0.52437122591 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 124.9933497873° = 124°59'37″ = 2.18215480815 rad

Height: ha = 5.40768330612
Height: hb = 6.39880857891
Height: hc = 3.30106486242

Median: ma = 9.40105214785
Median: mb = 10.07220841472
Median: mc = 3.37702574777

Inradius: r = 1.55222680016
Circumradius: R = 7.80884652244

Vertex coordinates: A[12.79436569493; 0] B[0; 0] C[7.07882638168; 3.30106486242]
Centroid: CG[6.62439735887; 1.11002162081]
Coordinates of the circumscribed circle: U[6.39768284747; -4.4788025751]
Coordinates of the inscribed circle: I[7.00218284747; 1.55222680016]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 149.9933497873° = 149°59'37″ = 0.52437122591 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 55.00765021269° = 55°23″ = 2.18215480815 rad


How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase is the calculation of other characteristics of the triangle, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines


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=7.81 b=6.6 c=12.79a = 7.81 \ \\ b = 6.6 \ \\ c = 12.79

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

p=a+b+c=7.81+6.6+12.79=27.2p = a+b+c = 7.81+6.6+12.79 = 27.2

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=p2=27.22=13.6s = \dfrac{ p }{ 2 } = \dfrac{ 27.2 }{ 2 } = 13.6

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides are known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T=s(sa)(sb)(sc) T=13.6(13.67.81)(13.66.6)(13.612.79) T=445.79=21.11T = \sqrt{ s(s-a)(s-b)(s-c) } \ \\ T = \sqrt{ 13.6(13.6-7.81)(13.6-6.6)(13.6-12.79) } \ \\ T = \sqrt{ 445.79 } = 21.11

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The area of a triangle is half of the product of the length of the base and the height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T=aha2  ha=2 Ta=2 21.117.81=5.41 hb=2 Tb=2 21.116.6=6.4 hc=2 Tc=2 21.1112.79=3.3T = \dfrac{ a h _a }{ 2 } \ \\ \ \\ h _a = \dfrac{ 2 \ T }{ a } = \dfrac{ 2 \cdot \ 21.11 }{ 7.81 } = 5.41 \ \\ h _b = \dfrac{ 2 \ T }{ b } = \dfrac{ 2 \cdot \ 21.11 }{ 6.6 } = 6.4 \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 21.11 }{ 12.79 } = 3.3

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding the angles of a triangle when we know all three sides. The cosine rule, also known as the law of cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines is the extrapolation of the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines, because cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from cosine value.

a2=b2+c22bccosα  α=arccos(b2+c2a22bc)=arccos(6.62+12.7927.8122 6.6 12.79)=3023"  b2=a2+c22accosβ β=arccos(a2+c2b22ac)=arccos(7.812+12.7926.622 7.81 12.79)=25 γ=180αβ=1803023"25=1245937"a^2 = b^2+c^2 - 2bc \cos α \ \\ \ \\ α = \arccos(\dfrac{ b^2+c^2-a^2 }{ 2bc } ) = \arccos(\dfrac{ 6.6^2+12.79^2-7.81^2 }{ 2 \cdot \ 6.6 \cdot \ 12.79 } ) = 30^\circ 23" \ \\ \ \\ b^2 = a^2+c^2 - 2ac \cos β \ \\ β = \arccos(\dfrac{ a^2+c^2-b^2 }{ 2ac } ) = \arccos(\dfrac{ 7.81^2+12.79^2-6.6^2 }{ 2 \cdot \ 7.81 \cdot \ 12.79 } ) = 25^\circ \ \\ γ = 180^\circ - α - β = 180^\circ - 30^\circ 23" - 25^\circ = 124^\circ 59'37"

7. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

T=rs r=Ts=21.1113.6=1.55T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 21.11 }{ 13.6 } = 1.55

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R=abc4 rs=7.81 6.6 12.794 1.552 13.602=7.81R = \dfrac{ a b c }{ 4 \ r s } = \dfrac{ 7.81 \cdot \ 6.6 \cdot \ 12.79 }{ 4 \cdot \ 1.552 \cdot \ 13.602 } = 7.81

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

ma=2b2+2c2a22=2 6.62+2 12.7927.8122=9.401 mb=2c2+2a2b22=2 12.792+2 7.8126.622=10.072 mc=2a2+2b2c22=2 7.812+2 6.6212.7922=3.37m_a = \dfrac{ \sqrt{ 2b^2+2c^2 - a^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 6.6^2+2 \cdot \ 12.79^2 - 7.81^2 } }{ 2 } = 9.401 \ \\ m_b = \dfrac{ \sqrt{ 2c^2+2a^2 - b^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 12.79^2+2 \cdot \ 7.81^2 - 6.6^2 } }{ 2 } = 10.072 \ \\ m_c = \dfrac{ \sqrt{ 2a^2+2b^2 - c^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 7.81^2+2 \cdot \ 6.6^2 - 12.79^2 } }{ 2 } = 3.37



#2 Obtuse scalene triangle.

Sides: a = 7.81   b = 6.6   c = 1.36328706842

Area: T = 2.24991786244
Perimeter: p = 15.77328706842
Semiperimeter: s = 7.88664353421

Angle ∠ A = α = 149.9933497873° = 149°59'37″ = 2.61878803945 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 5.00765021269° = 5°23″ = 0.08773799461 rad

Height: ha = 0.57659740395
Height: hb = 0.68215692801
Height: hc = 3.30106486242

Median: ma = 2.73112420711
Median: mb = 4.53217500208
Median: mc = 7.19881730928

Inradius: r = 0.28551958492
Circumradius: R = 7.80884652244

Vertex coordinates: A[1.36328706842; 0] B[0; 0] C[7.07882638168; 3.30106486242]
Centroid: CG[2.81437115003; 1.11002162081]
Coordinates of the circumscribed circle: U[0.68114353421; 7.77986743752]
Coordinates of the inscribed circle: I[1.28664353421; 0.28551958492]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 30.00765021269° = 30°23″ = 2.61878803945 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 174.9933497873° = 174°59'37″ = 0.08773799461 rad

Calculate another triangle

How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase is the calculation of other characteristics of the triangle, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines


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.

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

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=p2=15.772=7.89s = \dfrac{ p }{ 2 } = \dfrac{ 15.77 }{ 2 } = 7.89

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides are known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The area of a triangle is half of the product of the length of the base and the height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T=aha2  ha=2 Ta=2 2.257.81=0.58 hb=2 Tb=2 2.256.6=0.68 hc=2 Tc=2 2.251.36=3.3T = \dfrac{ a h _a }{ 2 } \ \\ \ \\ h _a = \dfrac{ 2 \ T }{ a } = \dfrac{ 2 \cdot \ 2.25 }{ 7.81 } = 0.58 \ \\ h _b = \dfrac{ 2 \ T }{ b } = \dfrac{ 2 \cdot \ 2.25 }{ 6.6 } = 0.68 \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 2.25 }{ 1.36 } = 3.3

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding the angles of a triangle when we know all three sides. The cosine rule, also known as the law of cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines is the extrapolation of the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines, because cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from cosine value.

7. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

ma=2b2+2c2a22=2 6.62+2 1.3627.8122=2.731 mb=2c2+2a2b22=2 1.362+2 7.8126.622=4.532 mc=2a2+2b2c22=2 7.812+2 6.621.3622=7.198m_a = \dfrac{ \sqrt{ 2b^2+2c^2 - a^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 6.6^2+2 \cdot \ 1.36^2 - 7.81^2 } }{ 2 } = 2.731 \ \\ m_b = \dfrac{ \sqrt{ 2c^2+2a^2 - b^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 1.36^2+2 \cdot \ 7.81^2 - 6.6^2 } }{ 2 } = 4.532 \ \\ m_c = \dfrac{ \sqrt{ 2a^2+2b^2 - c^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 7.81^2+2 \cdot \ 6.6^2 - 1.36^2 } }{ 2 } = 7.198

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