Right triangle calculator (a,b)

Please enter two properties of the right triangle

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


You have entered cathetus a and cathetus b.

Right isosceles triangle.

Sides: a = 32   b = 32   c = 45.25548339959

Area: T = 512
Perimeter: p = 109.2554833996
Semiperimeter: s = 54.6277416998

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 32
Height: hb = 32
Height: hc = 22.6277416998

Median: ma = 35.777708764
Median: mb = 35.777708764
Median: mc = 22.6277416998

Inradius: r = 9.3732583002
Circumradius: R = 22.6277416998

Vertex coordinates: A[45.25548339959; 0] B[0; 0] C[22.6277416998; 22.6277416998]
Centroid: CG[22.6277416998; 7.54224723327]
Coordinates of the circumscribed circle: U[22.6277416998; -0]
Coordinates of the inscribed circle: I[22.6277416998; 9.3732583002]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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. Input data entered: cathetus a and cathetus b

a=32 b=32a = 32 \ \\ b = 32

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

c2=a2+b2 c=a2+b2=322+322=2048=45.255c^2 = a^2+b^2 \ \\ c = \sqrt{ a^2+b^2 } = \sqrt{ 32^2 + 32^2 } = \sqrt{ 2048 } = 45.255

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=32 b=32 c=45.25a = 32 \ \\ b = 32 \ \\ c = 45.25

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

p=a+b+c=32+32+45.25=109.25p = a+b+c = 32+32+45.25 = 109.25

4. 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=109.252=54.63s = \dfrac{ p }{ 2 } = \dfrac{ 109.25 }{ 2 } = 54.63

5. The triangle area - from two legs

T=ab2=32⋅ 322=512T = \dfrac{ ab }{ 2 } = \dfrac{ 32 \cdot \ 32 }{ 2 } = 512

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

ha=b=32  hb=a=32  T=chc2   hc=2 Tc=2⋅ 51245.25=22.63h _a = b = 32 \ \\ \ \\ h _b = a = 32 \ \\ \ \\ T = \dfrac{ c h _c }{ 2 } \ \\ \ \\ \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 512 }{ 45.25 } = 22.63

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

sin⁡α=ac α=arcsin⁡(ac)=arcsin⁡(3245.25)=45∘ sin⁡β=bc β=arcsin⁡(bc)=arcsin⁡(3245.25)=45∘ γ=90∘\sin α = \dfrac{ a }{ c } \ \\ α = \arcsin(\dfrac{ a }{ c } ) = \arcsin(\dfrac{ 32 }{ 45.25 } ) = 45^\circ \ \\ \sin β = \dfrac{ b }{ c } \ \\ β = \arcsin(\dfrac{ b }{ c } ) = \arcsin(\dfrac{ 32 }{ 45.25 } ) = 45^\circ \ \\ γ = 90^\circ

8. 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=51254.63=9.37T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 512 }{ 54.63 } = 9.37

9. 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=c2=45.252=22.63R = \dfrac{ c }{ 2 } = \dfrac{ 45.25 }{ 2 } = 22.63

10. 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.

ma2=b2+(a/2)2 ma=b2+(a/2)2=322+(32/2)2=35.777  mb2=a2+(b/2)2 mb=a2+(b/2)2=322+(32/2)2=35.777  mc=R=c2=45.252=22.627m_a^2 = b^2 + (a/2)^2 \ \\ m_a = \sqrt{ b^2 + (a/2)^2 } = \sqrt{ 32^2 + (32/2)^2 } = 35.777 \ \\ \ \\ m_b^2 = a^2 + (b/2)^2 \ \\ m_b = \sqrt{ a^2 + (b/2)^2 } = \sqrt{ 32^2 + (32/2)^2 } = 35.777 \ \\ \ \\ m_c = R = \dfrac{ c }{ 2 } = \dfrac{ 45.25 }{ 2 } = 22.627

Calculate another triangle


The right triangle calculators compute angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in the real world. Two independent properties entirely determine any right-angled triangle. The calculator provides a step-by-step explanation for each calculation.

A right triangle is a kind of triangle that has one angle that measures C=90°. In a Right triangle, the side c that is opposite of the C=90° angle, is the longest side of the triangle and is called the hypotenuse. The variables a, b are the lengths of the shorter sides, also called legs or arms. Variables for angles are A, B, or α (alpha) and β (beta). Variable h refers to the altitude(height) of the triangle, which is the length from the vertex C to the hypotenuse of the triangle.

Examples for right triangle calculation:

A right triangle in word problems in mathematics:

  • Triangle P2
    1right_triangle Can triangle have two right angles?
  • The ditch
    prikop Ditch with cross section of an isosceles trapezoid with bases 2m 6m are deep 1.5m. How long is the slope of the ditch?
  • Height 2
    1unilateral_triangle Calculate the height of the equilateral triangle with side 38.
  • Vector 7
    vectors_sum0_1 Given vector OA(12,16) and vector OB(4,1). Find vector AB and vector |A|.
  • Area of RT 2
    tr Calculate the area of right triangle whose legs have a length 5.8 cm and 5.8 cm.
  • Cableway
    cable-car Cableway has a length of 1800 m. The horizontal distance between the upper and lower cable car station is 1600 m. Calculate how much meters altitude is higher upper station than the base station.
  • Right triangle
    right_triangle Right triangle legs has lengths 630 mm and 411 dm. Calculate the area of this triangle.
  • ABS CN
    complex_num Calculate the absolute value of complex number -15-29i.
  • Ladder
    rebrik_2 The ladder has a length 3.5 meters. He is leaning against the wall so that his bottom end is 2 meters away from the wall. Determine the height of the ladder.
  • Broken tree
    stromy_4 The tree is broken at 4 meters above the ground and the top of the tree touches the ground at a distance of 5 from the trunk. Calculate the original height of the tree.
  • If the
    tan If the tangent of an angle of a right angled triangle is 0.8. Then its longest side is. .. .
  • Right angled
    right-triangle From the right triangle with legs 12 cm and 20 cm we built a square with the same content as the triangle. How long will be side of the square?
  • Height
    thales_law Is right that in any right triangle height is less or equal half of the hypotenuse?
  • Cable car
    lanovka Cable car rises at an angle 45° and connects the upper and lower station with an altitude difference of 744 m. How long is "endless" tow rope?
  • Trapezoid - RR
    right_trapezium Find the area of the right angled trapezoid ABCD with the right angle at the A vertex; a = 3 dm b = 5 dm c = 6 dm d = 4 dm

next math problems »

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

See more information about triangles or more details on solving triangles.