# Right triangle calculator - the result

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R

You have entered hypotenuse c, and angle β.

### Right scalene triangle.

Sides: a = 7.8143635707   b = 22.69224458144   c = 24

Area: T = 88.65552524469
Perimeter: p = 54.50660815214
Semiperimeter: s = 27.25330407607

Angle ∠ A = α = 19° = 0.33216125579 rad
Angle ∠ B = β = 71° = 1.23991837689 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 22.69224458144
Height: hb = 7.8143635707
Height: hc = 7.38879377039

Median: ma = 23.02662963322
Median: mb = 13.77664174306
Median: mc = 12

Vertex coordinates: A[24; 0] B[0; 0] C[2.54438709567; 7.38879377039]
Centroid: CG[8.84879569856; 2.46326459013]
Coordinates of the circumscribed circle: U[12; 0]
Coordinates of the inscribed circle: I[4.56105949463; 3.25330407607]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 161° = 0.33216125579 rad
∠ B' = β' = 109° = 1.23991837689 rad
∠ C' = γ' = 90° = 1.57107963268 rad

# How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is to 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 calculates 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. Input data entered: hypotenuse c, and angle β

$c = 24 \ \\ β = 71^\circ$

### 2. From angle β we calculate angle α:

$α + β + 90^\circ = 180^\circ \ \\ α = 90^\circ - β = 90^\circ - 71 ^\circ = 19 ^\circ$

### 3. From hypotenuse c and angle α we calculate cathetus a:

$\sin α = a:c \ \\ a = c \cdot \ \sin α = 24 \cdot \ \sin(19 ^\circ ) = 7.814$

### 4. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

$c^2 = a^2+b^2 \ \\ b = \sqrt{ c^2 - a^2 } = \sqrt{ 24^2 - 7.814^2 } = 22.692$

Now we know the lengths of all three sides of the triangle, and the triangle is uniquely determined. Next, we calculate another of its characteristics - the same procedure as calculation of the triangle from the known three sides SSS.

$a = 7.81 \ \\ b = 22.69 \ \\ c = 24$

### 6. 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.

### 9. Calculation of the inner angles of the triangle - basic use of sine function

$\sin α = \dfrac{ a }{ c } \ \\ α = \arcsin(\dfrac{ a }{ c } ) = \arcsin(\dfrac{ 7.81 }{ 24 } ) = 19^\circ \ \\ \sin β = \dfrac{ b }{ c } \ \\ β = \arcsin(\dfrac{ b }{ c } ) = \arcsin(\dfrac{ 22.69 }{ 24 } ) = 71^\circ \ \\ γ = 90^\circ$

An incircle of a triangle is a circle which is tangent 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 the inradius and semiperimeter (half the perimeter) of a triangle is its area.

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.

$R = \dfrac{ c }{ 2 } = \dfrac{ 24 }{ 2 } = 12$

### 12. 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 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.