Isosceles triangle calculator (S)

You have entered Area S and Perimeter o.

Acute isosceles Triangle.

The lengths of the sides of the triangle:
a = 5
b = 5
c = 6

Area: T = 12
Perimeter: p = 16
Semiperimeter: s = 8

Angle ∠ A = α = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 73.74397952917° = 73°44'23″ = 1.28770022176 rad

Altitude to side a: h_a = 4.8
Altitude to side b: h_b = 4.8
Altitude to side c: h_c = 4

Median: m_a = 4.92444289009
Median: m_b = 4.92444289009
Median: m_c = 4

Inradius: r = 1.5
Circumradius: R = 3.125

Vertex coordinates: A[6; 0] B[0; 0] C[3; 4]
Centroid: CG[3; 1.33333333333]
Coordinates of the circumscribed circle: U[3; 0.875]
Coordinates of the inscribed circle: I[3; 1.5]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 126.87698976458° = 126°52'12″ = 2.21442974356 rad
∠ B' = β' = 126.87698976458° = 126°52'12″ = 2.21442974356 rad
∠ C' = γ' = 106.26602047083° = 106°15'37″ = 1.8554590436 rad

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How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other triangle characteristics, 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. Input data entered: Area S and Perimeter o

S = 12 o = 16

2. From the Area S and Perimeter o, we calculate side c:

T = \frac{c h}{2} 2 T = c h 2 a + c = o c = o - 2 a 2 o c^{3} - o^{2} c^{2} + 16 T^{2} = 0 2 \cdot 16 c^{3} - 1 6^{2} c^{2} + 161 2^{2} = 0 32 c^{3} - 256 c^{2} + 2304 = 0 c_{1} = 6 c_{2} = - 2.606 c_{3} = 4.606 c = 6

3. From the Perimeter o and side c, we calculate Side a:

o = 2 a + c a = (o - c) / 2 = (16 - 6) / 2 = 5

4. From the Side a, we calculate Side b:

b = a = 5

5. From the Area S and side c, we calculate Altitude h_c:

T = \frac{c h}{2} h = \frac{2 \cdot T}{c} = \frac{2 \cdot 1 2}{6} = 4

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides (SSS).

a = 5 b = 5 c = 6

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

p = a + b + c = 5 + 5 + 6 = 16

7. The semiperimeter of the triangle

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

s = \frac{p}{2} = \frac{1 6}{2} = 8

8. Calculate the height of the isosceles triangle.

9. The triangle area

10. Calculation of the inner angles of the triangle - symmetry

11. Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an 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 a triangle's inradius and semiperimeter (half the perimeter) is its area.

T = r s r = \frac{T}{s} = \frac{1 2}{8} = 1.5

12. 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. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

13. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. 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 of 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 a median's length from its side's lengths.

m_{a} = \frac{2 b ^{2} + 2 c ^{2} - a ^{2}}{2} = \frac{2 \cdot 5 ^{2} + 2 \cdot 6 ^{2} - 5 ^{2}}{2} = 4.924 m_{b} = \frac{2 c ^{2} + 2 a ^{2} - b ^{2}}{2} = \frac{2 \cdot 6 ^{2} + 2 \cdot 5 ^{2} - 5 ^{2}}{2} = 4.924 m_{c} = \frac{2 a ^{2} + 2 b ^{2} - c ^{2}}{2} = \frac{2 \cdot 5 ^{2} + 2 \cdot 5 ^{2} - 6 ^{2}}{2} = 4

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