Types of Triangles Definitions and Examples

Types of Triangles Definitions, Formulas, & Examples

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    Types of Triangles Definitions and Examples

    Introduction

    Whether we’re drawing them on a whiteboard to illustrate a point, or we’re seeing them in our classrooms, triangles are an important part of our lives. But what do these shapes actually mean? And what do they have to do with mathematics? In this blog post, we will explore the different types of triangles and provide examples. We will also discuss how triangles are used in mathematics, and how they can be helpful in problem solving.

    What are the Different Types of Triangles?

    Triangles are shapes that are made up of three lines that are either intersecting or parallel to each other. In terms of types, there are six different types of triangles: isosceles, equilateral, scalene, right angled triangle, acute angle triangle, and obtuse angle triangle.

    Isosceles triangles are the simplest type and consist of two equal sides and a vertex at the center that is the same height as the two other sides. The hypotenuse is also equal to the length of the shorter side.

    Equilateral triangles have uniform angles between all three sides. Each vertex is exactly 90 degrees from another one.

    Scalene triangles have at least one angle that is not a right angle. The other angles must be greater than 90 degrees but less than 180 degrees. This makes scalene triangles unique in that they can have more than six different configurations (four if you don’t count a 45-degree angle).

    Right angled triangles have one side that is perpendicular to the other two and has a 90-degree angle at the vertex.

    Acute angle triangles have an acute vertex (pointy end) and all three angles are less than 180 degrees.

    Obtuse angle triangles have an obtuse vertex (rounded end) and all three angles are greater than 180 degrees.

    Classifying Triangles

    Triangles can be classified in a variety of ways, depending on the angle measures used. One common way to classify triangles is by their base angles, which are measured from the corner to the point where the shortest side meets the other two sides.

    Another way to classify triangles is by their height. Triangles that have identical base angles but different heights are called “anhedral” or “isosceles.” Triangles that have identical height but different base angles are called “tetrahedral.”

    The third way to classify triangles is by their angles at the vertices. A triangle with equal angles at its vertices is called “right,” while a triangle with one angle greater than 90 degrees is called “left.”

    Types of Triangles Based on Sides

    Types of Triangles Based on Sides

    There are three basic types of triangles based on the sides: isosceles, equilateral, and scalene.

    Isosceles Triangles: Two of the triangle’s sides are the same length (equal in size). The triangle is isosceles if its base is exactly in the middle of one of the other two sides.

    Equilateral Triangles: All three sides are equal in length. The triangle is equilateral if it has its three vertices at the same location and at right angles to each other.

    Scalene Triangles: One side is longer than the other two – this side is called the Scale Side. The other two sides are called the Base Sides. A scalene triangle has one vertex on its Scale Side and the remaining two vertices on its Base Sides.

    Types of Triangles Based on Angles

    Triangles are one of the most basic shapes in geometry. In this post, we will explore three different types of triangles based on angles: right triangles, supplementary angles, and congruent triangles.

    Right Triangles
    A right triangle is composed of a hypotenuse angle (the smallest angle measuring between the two other angles), called the right angle, and two other angles that are measured from the right angle. The sum of these three angles is 180 degrees. All right triangles have a unique length ratio: the short side is always shorter than the long side. Right triangle examples include those found in standard geometry textbooks and on test questions.

    Supplementary Angles
    In a supplementary angle, one angle is twice the other angle. Supplementary angles are found in pairs along with their associated right triangle properties: each supplementary angle has a corresponding opposite Supplementary angle and both sides are equal by ratio. For example, if someone flips an object so that its long side becomes its shortest side, then that object would now be considered to have two supplementary angles-the acute (or concave)angle at the top and bottom of the flipped object, and the obtuse (or convex)angle formed by extending one of the legs down from where it meets the front of the flipped object perpendicular to its original length

    Types of Triangle Based on Sides and Angles

    There are many types of triangles, based on the sides and angles of the triangle. Here are a few examples:

    Perpendicular Triangle: The base angles are both 180 degrees.

    Right Triangle: The two angles at the right side are 90 degrees.

    Inverse Square Triangle: The three angles in a triangle add up to 180 degrees.

    Isosceles Triangle: Two of the sides are equal in length, and the third side is equal to the two other side’s combined length.

    Conclusion

    In this article, we will be exploring the different types of triangles and their definitions. After reading this, you should have a better understanding of what these shapes are, as well as an idea of how to create them using basic geometry laws. So without further ado, let’s get started!

    Types of Triangles

    Result

    30-60-90 triangle | convex laminae | polygonal laminae | right triangular laminae | scalene triangular laminae | triangular laminae 3, 4, 5 triangle | convex laminae | polygonal laminae | right triangular laminae | scalene triangular laminae | triangular laminae equilateral triangle | convex laminae | equilateral polygonal laminae | polygonal laminae | regular polygonal laminae | triangular laminae inverted equilateral triangle | convex laminae | equilateral polygonal laminae | polygonal laminae | regular polygonal laminae | triangular laminae golden triangle | convex laminae | isosceles triangular laminae | polygonal laminae | triangular laminae isosceles right triangle | convex laminae | isosceles triangular laminae | polygonal laminae | right triangular laminae | triangular laminae isosceles triangle | convex laminae | isosceles triangular laminae | polygonal laminae | triangular laminae Kepler triangle | convex laminae | polygonal laminae | right triangular laminae | scalene triangular laminae | triangular laminae pennant | convex laminae | isosceles triangular laminae | polygonal laminae | triangular laminae right triangle | convex laminae | polygonal laminae | right triangular laminae | triangular laminae

    Definitions

    Definitions 30-60-90 triangle

    Definitions 3, 4, 5 triangle

    Definitions Equilateral triangle

    Definitions 30-60-90 triangle

    Definitions 3, 4, 5 triangle

    Definitions Equilateral triangle

    Defining inequalities

    y>=0 and sqrt(3) a>=2 (sqrt(3) x + y) and x>=0

    y>=0 and 12 a>=3 x + 4 y and x>=0

    sqrt(3) (a + 3 x)>=3 y and sqrt(3) a + 6 y>=0 and sqrt(3) a>=3 (sqrt(3) x + y)

    sqrt(3) (a + 3 x)>=-3 y and sqrt(3) a>=6 y and sqrt(3) a + 3 y>=3 sqrt(3) x

    -a/2<=x<=a/2 and 0<=y<=1/2 sqrt(5 + 2 sqrt(5)) a (1 - 2 abs(x/a))

    y>=0 and a>=x + y and x>=0

    -a/2<=x<=a/2 and 0<=y<=h (1 - 2 abs(x/a))

    y>=0 and a>=x/sqrt(ϕ) + y and x>=0

    h (a + 2 y)>=a x and a h>=a x + 2 h y and x>=0

    y>=0 and a b>=a y + b x and x>=0

    y>=0 and sqrt(3) a>=2 (sqrt(3) x + y) and x>=0

    y>=0 and 12 a>=3 x + 4 y and x>=0

    sqrt(3) (a + 3 x)>=3 y and sqrt(3) a + 6 y>=0 and sqrt(3) a>=3 (sqrt(3) x + y)

    sqrt(3) (a + 3 x)>=-3 y and sqrt(3) a>=6 y and sqrt(3) a + 3 y>=3 sqrt(3) x

    -a/2<=x<=a/2 and 0<=y<=1/2 sqrt(5 + 2 sqrt(5)) a (1 - 2 abs(x/a))

    y>=0 and a>=x + y and x>=0

    -a/2<=x<=a/2 and 0<=y<=h (1 - 2 abs(x/a))

    y>=0 and a>=x/sqrt(ϕ) + y and x>=0

    h (a + 2 y)>=a x and a h>=a x + 2 h y and x>=0

    y>=0 and a b>=a y + b x and x>=0

    Alternate names

    sublime triangle

    horizontal isosceles triangle

    scalene trigon

    trigon

    Lamina properties

    30-60-90 triangle | (0, 0) | (a/2, 0) | (0, (sqrt(3) a)/2) 3, 4, 5 triangle | (0, 3 a) | (0, 0) | (4 a, 0) equilateral triangle | (0, a/sqrt(3)) | (-a/2, -a/(2 sqrt(3))) | (a/2, -a/(2 sqrt(3))) inverted equilateral triangle | (0, -a/sqrt(3)) | (a/2, a/(2 sqrt(3))) | (-a/2, a/(2 sqrt(3))) golden triangle | (-a/2, 0) | (a/2, 0) | (0, 1/2 sqrt(5 + 2 sqrt(5)) a) isosceles right triangle | (0, a) | (0, 0) | (a, 0) isosceles triangle | (-a/2, 0) | (a/2, 0) | (0, h) Kepler triangle | (0, 0) | (sqrt(ϕ) a, 0) | (0, a) pennant | (0, a/2) | (0, -a/2) | (h, 0) right triangle | (0, 0) | (a, 0) | (0, b) scalene triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0) triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

    30-60-90 triangle | 3 3, 4, 5 triangle | 3 equilateral triangle | 3 inverted equilateral triangle | 3 golden triangle | 3 isosceles right triangle | 3 isosceles triangle | 3 Kepler triangle | 3 pennant | 3 right triangle | 3 scalene triangle | 3 triangle | 3

    30-60-90 triangle | a>0 3, 4, 5 triangle | a>0 equilateral triangle | a>0 inverted equilateral triangle | a>0 golden triangle | a>0 isosceles right triangle | a>0 isosceles triangle | a>0 and h>0 Kepler triangle | a>0 pennant | a>0 and h>0 right triangle | a>0 and b>0 scalene triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b and a!=b!=c triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

    equilateral triangle | r = a/(2 sqrt(3)) inverted equilateral triangle | r = a/(2 sqrt(3))

    equilateral triangle | h = a/(2 sqrt(3)) inverted equilateral triangle | h = a/(2 sqrt(3))

    30-60-90 triangle | (sqrt(3) a)/2 3, 4, 5 triangle | 3 a equilateral triangle | (sqrt(3) a)/2 inverted equilateral triangle | (sqrt(3) a)/2 golden triangle | 1/2 sqrt(5 + 2 sqrt(5)) a isosceles right triangle | a isosceles triangle | h Kepler triangle | a pennant | a right triangle | b scalene triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c) triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

    30-60-90 triangle | A = (sqrt(3) a^2)/8 3, 4, 5 triangle | A = 6 a^2 equilateral triangle | A = (sqrt(3) a^2)/4 inverted equilateral triangle | A = (sqrt(3) a^2)/4 golden triangle | A = 1/4 sqrt(5 + 2 sqrt(5)) a^2 isosceles right triangle | A = a^2/2 isosceles triangle | A = (a h)/2 Kepler triangle | A = (a^2 sqrt(ϕ))/2 pennant | A = (a h)/2 right triangle | A = (a b)/2 scalene triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    30-60-90 triangle | x^_ = (a/6, a/(2 sqrt(3))) 3, 4, 5 triangle | x^_ = ((4 a)/3, a) equilateral triangle | x^_ = (0, 0) inverted equilateral triangle | x^_ = (0, 0) golden triangle | x^_ = (0, 1/6 sqrt(5 + 2 sqrt(5)) a) isosceles right triangle | x^_ = (a/3, a/3) isosceles triangle | x^_ = (0, h/3) Kepler triangle | x^_ = ((sqrt(ϕ) a)/3, a/3) pennant | x^_ = (h/3, 0) right triangle | x^_ = (a/3, b/3) scalene triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c)) triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

    30-60-90 triangle | (0, 0) | (a/2, 0) | (0, (sqrt(3) a)/2) 3, 4, 5 triangle | (0, 3 a) | (0, 0) | (4 a, 0) equilateral triangle | (0, a/sqrt(3)) | (-a/2, -a/(2 sqrt(3))) | (a/2, -a/(2 sqrt(3))) inverted equilateral triangle | (0, -a/sqrt(3)) | (a/2, a/(2 sqrt(3))) | (-a/2, a/(2 sqrt(3))) golden triangle | (-a/2, 0) | (a/2, 0) | (0, 1/2 sqrt(5 + 2 sqrt(5)) a) isosceles right triangle | (0, a) | (0, 0) | (a, 0) isosceles triangle | (-a/2, 0) | (a/2, 0) | (0, h) Kepler triangle | (0, 0) | (sqrt(ϕ) a, 0) | (0, a) pennant | (0, a/2) | (0, -a/2) | (h, 0) right triangle | (0, 0) | (a, 0) | (0, b) scalene triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0) triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

    30-60-90 triangle | 3 3, 4, 5 triangle | 3 equilateral triangle | 3 inverted equilateral triangle | 3 golden triangle | 3 isosceles right triangle | 3 isosceles triangle | 3 Kepler triangle | 3 pennant | 3 right triangle | 3 scalene triangle | 3 triangle | 3

    30-60-90 triangle | a>0 3, 4, 5 triangle | a>0 equilateral triangle | a>0 inverted equilateral triangle | a>0 golden triangle | a>0 isosceles right triangle | a>0 isosceles triangle | a>0 and h>0 Kepler triangle | a>0 pennant | a>0 and h>0 right triangle | a>0 and b>0 scalene triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b and a!=b!=c triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

    equilateral triangle | r = a/(2 sqrt(3)) inverted equilateral triangle | r = a/(2 sqrt(3))

    equilateral triangle | h = a/(2 sqrt(3)) inverted equilateral triangle | h = a/(2 sqrt(3))

    30-60-90 triangle | (sqrt(3) a)/2 3, 4, 5 triangle | 3 a equilateral triangle | (sqrt(3) a)/2 inverted equilateral triangle | (sqrt(3) a)/2 golden triangle | 1/2 sqrt(5 + 2 sqrt(5)) a isosceles right triangle | a isosceles triangle | h Kepler triangle | a pennant | a right triangle | b scalene triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c) triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

    30-60-90 triangle | A = (sqrt(3) a^2)/8 3, 4, 5 triangle | A = 6 a^2 equilateral triangle | A = (sqrt(3) a^2)/4 inverted equilateral triangle | A = (sqrt(3) a^2)/4 golden triangle | A = 1/4 sqrt(5 + 2 sqrt(5)) a^2 isosceles right triangle | A = a^2/2 isosceles triangle | A = (a h)/2 Kepler triangle | A = (a^2 sqrt(ϕ))/2 pennant | A = (a h)/2 right triangle | A = (a b)/2 scalene triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    30-60-90 triangle | x^_ = (a/6, a/(2 sqrt(3))) 3, 4, 5 triangle | x^_ = ((4 a)/3, a) equilateral triangle | x^_ = (0, 0) inverted equilateral triangle | x^_ = (0, 0) golden triangle | x^_ = (0, 1/6 sqrt(5 + 2 sqrt(5)) a) isosceles right triangle | x^_ = (a/3, a/3) isosceles triangle | x^_ = (0, h/3) Kepler triangle | x^_ = ((sqrt(ϕ) a)/3, a/3) pennant | x^_ = (h/3, 0) right triangle | x^_ = (a/3, b/3) scalene triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c)) triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

    Mechanical properties

    30-60-90 triangle | J_x invisible comma x = (sqrt(3) a^4)/64 3, 4, 5 triangle | J_x invisible comma x = 9 a^4 equilateral triangle | J_x invisible comma x = a^4/(32 sqrt(3)) inverted equilateral triangle | J_x invisible comma x = a^4/(32 sqrt(3)) golden triangle | J_x invisible comma x = 1/96 (5 + 2 sqrt(5))^(3/2) a^4 isosceles right triangle | J_x invisible comma x = a^4/12 isosceles triangle | J_x invisible comma x = (a h^3)/12 Kepler triangle | J_x invisible comma x = (a^4 sqrt(ϕ))/12 pennant | J_x invisible comma x = (a^3 h)/48 right triangle | J_x invisible comma x = (a b^3)/12 scalene triangle | J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2) triangle | J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

    30-60-90 triangle | J_y invisible comma y = a^4/(64 sqrt(3)) 3, 4, 5 triangle | J_y invisible comma y = 16 a^4 equilateral triangle | J_y invisible comma y = a^4/(32 sqrt(3)) inverted equilateral triangle | J_y invisible comma y = a^4/(32 sqrt(3)) golden triangle | J_y invisible comma y = 1/96 sqrt(5 + 2 sqrt(5)) a^4 isosceles right triangle | J_y invisible comma y = a^4/12 isosceles triangle | J_y invisible comma y = (a^3 h)/48 Kepler triangle | J_y invisible comma y = 1/12 a^4 ϕ^(3/2) pennant | J_y invisible comma y = (a h^3)/12 right triangle | J_y invisible comma y = (a^3 b)/12 scalene triangle | J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2) triangle | J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

    30-60-90 triangle | J_zz = a^4/(16 sqrt(3)) 3, 4, 5 triangle | J_zz = 25 a^4 equilateral triangle | J_zz = a^4/(16 sqrt(3)) inverted equilateral triangle | J_zz = a^4/(16 sqrt(3)) golden triangle | J_zz = 1/48 (3 + sqrt(5)) sqrt(5 + 2 sqrt(5)) a^4 isosceles right triangle | J_zz = a^4/6 isosceles triangle | J_zz = 1/48 a h (a^2 + 4 h^2) Kepler triangle | J_zz = 1/12 a^4 ϕ^(3/2) + (a^4 sqrt(ϕ))/12 pennant | J_zz = 1/48 a h (a^2 + 4 h^2) right triangle | J_zz = 1/12 a b (a^2 + b^2) scalene triangle | J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2)) triangle | J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

    30-60-90 triangle | J_x invisible comma y = -a^4/128 3, 4, 5 triangle | J_x invisible comma y = -6 a^4 equilateral triangle | J_x invisible comma y = 0 inverted equilateral triangle | J_x invisible comma y = 0 golden triangle | J_x invisible comma y = 0 isosceles right triangle | J_x invisible comma y = -a^4/24 isosceles triangle | J_x invisible comma y = 0 Kepler triangle | J_x invisible comma y = -(a^4 ϕ)/24 pennant | J_x invisible comma y = 0 right triangle | J_x invisible comma y = -1/24 a^2 b^2 scalene triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2) triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

    30-60-90 triangle | r_x = a/(2 sqrt(2)) r_y = a/(2 sqrt(6)) 3, 4, 5 triangle | r_x = sqrt(3/2) a r_y = 2 sqrt(2/3) a equilateral triangle | r_x = a/(2 sqrt(6)) r_y = a/(2 sqrt(6)) inverted equilateral triangle | r_x = a/(2 sqrt(6)) r_y = a/(2 sqrt(6)) golden triangle | r_x = sqrt(5/24 + sqrt(5)/12) a r_y = a/(2 sqrt(6)) isosceles right triangle | r_x = a/sqrt(6) r_y = a/sqrt(6) isosceles triangle | r_x = h/sqrt(6) r_y = a/(2 sqrt(6)) Kepler triangle | r_x = a/sqrt(6) r_y = a sqrt(ϕ/6) pennant | r_x = a/(2 sqrt(6)) r_y = h/sqrt(6) right triangle | r_x = b/sqrt(6) r_y = a/sqrt(6) scalene triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)) triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

    equilateral triangle | K = (sqrt(3) a^4)/80 inverted equilateral triangle | K = (sqrt(3) a^4)/80 isosceles right triangle | K = a^4 (1/12 - (16 sum_n^∞ coth(1/2 π (2 n - 1))/(2 n - 1)^5)/π^5)

    30-60-90 triangle | J_x invisible comma x = (sqrt(3) a^4)/64 3, 4, 5 triangle | J_x invisible comma x = 9 a^4 equilateral triangle | J_x invisible comma x = a^4/(32 sqrt(3)) inverted equilateral triangle | J_x invisible comma x = a^4/(32 sqrt(3)) golden triangle | J_x invisible comma x = 1/96 (5 + 2 sqrt(5))^(3/2) a^4 isosceles right triangle | J_x invisible comma x = a^4/12 isosceles triangle | J_x invisible comma x = (a h^3)/12 Kepler triangle | J_x invisible comma x = (a^4 sqrt(ϕ))/12 pennant | J_x invisible comma x = (a^3 h)/48 right triangle | J_x invisible comma x = (a b^3)/12 scalene triangle | J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2) triangle | J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

    30-60-90 triangle | J_y invisible comma y = a^4/(64 sqrt(3)) 3, 4, 5 triangle | J_y invisible comma y = 16 a^4 equilateral triangle | J_y invisible comma y = a^4/(32 sqrt(3)) inverted equilateral triangle | J_y invisible comma y = a^4/(32 sqrt(3)) golden triangle | J_y invisible comma y = 1/96 sqrt(5 + 2 sqrt(5)) a^4 isosceles right triangle | J_y invisible comma y = a^4/12 isosceles triangle | J_y invisible comma y = (a^3 h)/48 Kepler triangle | J_y invisible comma y = 1/12 a^4 ϕ^(3/2) pennant | J_y invisible comma y = (a h^3)/12 right triangle | J_y invisible comma y = (a^3 b)/12 scalene triangle | J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2) triangle | J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

    30-60-90 triangle | J_zz = a^4/(16 sqrt(3)) 3, 4, 5 triangle | J_zz = 25 a^4 equilateral triangle | J_zz = a^4/(16 sqrt(3)) inverted equilateral triangle | J_zz = a^4/(16 sqrt(3)) golden triangle | J_zz = 1/48 (3 + sqrt(5)) sqrt(5 + 2 sqrt(5)) a^4 isosceles right triangle | J_zz = a^4/6 isosceles triangle | J_zz = 1/48 a h (a^2 + 4 h^2) Kepler triangle | J_zz = 1/12 a^4 ϕ^(3/2) + (a^4 sqrt(ϕ))/12 pennant | J_zz = 1/48 a h (a^2 + 4 h^2) right triangle | J_zz = 1/12 a b (a^2 + b^2) scalene triangle | J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2)) triangle | J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

    30-60-90 triangle | J_x invisible comma y = -a^4/128 3, 4, 5 triangle | J_x invisible comma y = -6 a^4 equilateral triangle | J_x invisible comma y = 0 inverted equilateral triangle | J_x invisible comma y = 0 golden triangle | J_x invisible comma y = 0 isosceles right triangle | J_x invisible comma y = -a^4/24 isosceles triangle | J_x invisible comma y = 0 Kepler triangle | J_x invisible comma y = -(a^4 ϕ)/24 pennant | J_x invisible comma y = 0 right triangle | J_x invisible comma y = -1/24 a^2 b^2 scalene triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2) triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

    30-60-90 triangle | r_x = a/(2 sqrt(2)) r_y = a/(2 sqrt(6)) 3, 4, 5 triangle | r_x = sqrt(3/2) a r_y = 2 sqrt(2/3) a equilateral triangle | r_x = a/(2 sqrt(6)) r_y = a/(2 sqrt(6)) inverted equilateral triangle | r_x = a/(2 sqrt(6)) r_y = a/(2 sqrt(6)) golden triangle | r_x = sqrt(5/24 + sqrt(5)/12) a r_y = a/(2 sqrt(6)) isosceles right triangle | r_x = a/sqrt(6) r_y = a/sqrt(6) isosceles triangle | r_x = h/sqrt(6) r_y = a/(2 sqrt(6)) Kepler triangle | r_x = a/sqrt(6) r_y = a sqrt(ϕ/6) pennant | r_x = a/(2 sqrt(6)) r_y = h/sqrt(6) right triangle | r_x = b/sqrt(6) r_y = a/sqrt(6) scalene triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)) triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

    equilateral triangle | K = (sqrt(3) a^4)/80 inverted equilateral triangle | K = (sqrt(3) a^4)/80 isosceles right triangle | K = a^4 (1/12 - (16 sum_n^∞ coth(1/2 π (2 n - 1))/(2 n - 1)^5)/π^5)

    Distance properties

    30-60-90 triangle | a/2 | a | (sqrt(3) a)/2 3, 4, 5 triangle | 3 a | 4 a | 5 a equilateral triangle | a | a | a inverted equilateral triangle | a | a | a golden triangle | a | a ϕ | a ϕ isosceles right triangle | a | a | sqrt(2) a isosceles triangle | a | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2) Kepler triangle | a sqrt(ϕ) | a ϕ | a pennant | a | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2) right triangle | a | sqrt(a^2 + b^2) | b scalene triangle | a | b | c triangle | a | b | c

    30-60-90 triangle | a 3, 4, 5 triangle | 5 a isosceles right triangle | sqrt(2) a Kepler triangle | a ϕ right triangle | sqrt(a^2 + b^2)

    30-60-90 triangle | p = 1/2 (3 + sqrt(3)) a 3, 4, 5 triangle | p = 12 a equilateral triangle | p = 3 a inverted equilateral triangle | p = 3 a golden triangle | p = a (2 ϕ + 1) isosceles right triangle | p = (2 + sqrt(2)) a isosceles triangle | p = sqrt(a^2 + 4 h^2) + a Kepler triangle | p = a (sqrt(ϕ) + ϕ + 1) pennant | p = sqrt(a^2 + 4 h^2) + a right triangle | p = sqrt(a^2 + b^2) + a + b scalene triangle | p = a + b + c triangle | p = a + b + c

    30-60-90 triangle | r = 1/4 (sqrt(3) - 1) a 3, 4, 5 triangle | r = a equilateral triangle | r = a/(2 sqrt(3)) inverted equilateral triangle | r = a/(2 sqrt(3)) golden triangle | r = 1/2 sqrt(5 - 2 sqrt(5)) a isosceles right triangle | r = a - a/sqrt(2) isosceles triangle | r = (a (sqrt(a^2 + 4 h^2) - a))/(4 h) Kepler triangle | r = 1/2 a (ϕ - 1) (ϕ^(3/2) - 1) pennant | r = (a (sqrt(a^2 + 4 h^2) - a))/(4 h) right triangle | r = 1/2 (-sqrt(a^2 + b^2) + a + b) scalene triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c)) triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

    30-60-90 triangle | R = a/2 3, 4, 5 triangle | R = (5 a)/2 equilateral triangle | R = a/sqrt(3) inverted equilateral triangle | R = a/sqrt(3) golden triangle | R = sqrt(1/10 (5 + sqrt(5))) a isosceles right triangle | R = a/sqrt(2) isosceles triangle | R = 1/8 (a^2/h + 4 h) Kepler triangle | R = (a ϕ)/2 pennant | R = 1/8 (a^2/h + 4 h) right triangle | R = 1/2 sqrt(a^2 + b^2) scalene triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    30-60-90 triangle | a 3, 4, 5 triangle | 5 a equilateral triangle | a inverted equilateral triangle | a golden triangle | a ϕ isosceles right triangle | sqrt(2) a isosceles triangle | max(a, sqrt(a^2/4 + h^2)) Kepler triangle | a ϕ pennant | max(a, sqrt(a^2/4 + h^2)) right triangle | sqrt(a^2 + b^2) scalene triangle | max(a, b, c) triangle | max(a, b, c)

    30-60-90 triangle | χ = 1 3, 4, 5 triangle | χ = 1 equilateral triangle | χ = 1 inverted equilateral triangle | χ = 1 golden triangle | χ = 1 isosceles right triangle | χ = 1 isosceles triangle | χ = 1 Kepler triangle | χ = 1 pennant | χ = 1 right triangle | χ = 1 scalene triangle | χ = 1 triangle | χ = 1

    30-60-90 triangle | s^_ = (a (204 + 36 sqrt(3) + 81 log(3) + 2 (9 + 8 sqrt(3)) log(2 + sqrt(3))))/1440 3, 4, 5 triangle | s^_ = (a (20460 + 9728 log(2) + 5103 log(3)))/22500 equilateral triangle | s^_ = 1/20 a (3 + 3 log(3)) inverted equilateral triangle | s^_ = 1/20 a (3 + 3 log(3)) golden triangle | s^_ = 1/60 a ((5 + 2 sqrt(5)) log(1 + 2/sqrt(5)) + 2 (7 + sqrt(5) + sqrt(5) sinh^(-1)(2))) isosceles right triangle | s^_ = 1/30 a (2 + 4 sqrt(2) + (4 + sqrt(2)) sinh^(-1)(1)) isosceles triangle | s^_ = (-a^5 + 8 a^3 h^2 + 8 a^4 sqrt(a^2 + 4 h^2) + 2 h^2 ((a^2 + 4 h^2)^(3/2) log((a (sqrt(a^2 + 4 h^2) + a))/(2 h^2) + 1) + 16 a^3 log((sqrt(a^2 + 4 h^2) + a)/a)) + 48 a h^4)/(30 a (a^2 + 4 h^2)^(3/2)) Kepler triangle | s^_ = (8 a (7 sqrt(2) + 3 sqrt(10) + 25 sqrt(1 + sqrt(5)) + 11 sqrt(5 (1 + sqrt(5))) + sqrt(2) (7 + 3 sqrt(5)) log(1/2 (1 + sqrt(5) + sqrt(2 (1 + sqrt(5))))) + sqrt(11 + 5 sqrt(5)) log(9 + 4 sqrt(5) + 2 sqrt(38 + 17 sqrt(5))) + sqrt(199 + 89 sqrt(5)) sinh^(-1)(2)))/(15 (1 + sqrt(5))^(9/2)) right triangle | s^_ = (2 a^5 b + 4 a^3 b^3 + a b^4 (sqrt(a^2 + b^2) + 2 b) + a^4 b sqrt(a^2 + b^2) + (a^2 + b^2)^(3/2) (b^3 log((sqrt(a^2 + b^2) + a)/b) + a^3 log((sqrt(a^2 + b^2) + b)/a)) + 2 a^3 b^3 coth^(-1)((a + b)/sqrt(a^2 + b^2)))/(15 a b (a^2 + b^2)^(3/2)) scalene triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c) triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)

    3, 4, 5 triangle | A^_ = a^2/2 equilateral triangle | A^_ = a^2/(16 sqrt(3)) inverted equilateral triangle | A^_ = a^2/(16 sqrt(3)) golden triangle | A^_ = 1/48 sqrt(5 + 2 sqrt(5)) a^2 isosceles right triangle | A^_ = a^2/24 isosceles triangle | A^_ = (a h)/24 Kepler triangle | A^_ = (a^2 sqrt(ϕ))/24 pennant | A^_ = (a h)/24 right triangle | A^_ = (a b)/24 scalene triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    30-60-90 triangle | a/2 | a | (sqrt(3) a)/2 3, 4, 5 triangle | 3 a | 4 a | 5 a equilateral triangle | a | a | a inverted equilateral triangle | a | a | a golden triangle | a | a ϕ | a ϕ isosceles right triangle | a | a | sqrt(2) a isosceles triangle | a | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2) Kepler triangle | a sqrt(ϕ) | a ϕ | a pennant | a | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2) right triangle | a | sqrt(a^2 + b^2) | b scalene triangle | a | b | c triangle | a | b | c

    30-60-90 triangle | a 3, 4, 5 triangle | 5 a isosceles right triangle | sqrt(2) a Kepler triangle | a ϕ right triangle | sqrt(a^2 + b^2)

    30-60-90 triangle | p = 1/2 (3 + sqrt(3)) a 3, 4, 5 triangle | p = 12 a equilateral triangle | p = 3 a inverted equilateral triangle | p = 3 a golden triangle | p = a (2 ϕ + 1) isosceles right triangle | p = (2 + sqrt(2)) a isosceles triangle | p = sqrt(a^2 + 4 h^2) + a Kepler triangle | p = a (sqrt(ϕ) + ϕ + 1) pennant | p = sqrt(a^2 + 4 h^2) + a right triangle | p = sqrt(a^2 + b^2) + a + b scalene triangle | p = a + b + c triangle | p = a + b + c

    30-60-90 triangle | r = 1/4 (sqrt(3) - 1) a 3, 4, 5 triangle | r = a equilateral triangle | r = a/(2 sqrt(3)) inverted equilateral triangle | r = a/(2 sqrt(3)) golden triangle | r = 1/2 sqrt(5 - 2 sqrt(5)) a isosceles right triangle | r = a - a/sqrt(2) isosceles triangle | r = (a (sqrt(a^2 + 4 h^2) - a))/(4 h) Kepler triangle | r = 1/2 a (ϕ - 1) (ϕ^(3/2) - 1) pennant | r = (a (sqrt(a^2 + 4 h^2) - a))/(4 h) right triangle | r = 1/2 (-sqrt(a^2 + b^2) + a + b) scalene triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c)) triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

    30-60-90 triangle | R = a/2 3, 4, 5 triangle | R = (5 a)/2 equilateral triangle | R = a/sqrt(3) inverted equilateral triangle | R = a/sqrt(3) golden triangle | R = sqrt(1/10 (5 + sqrt(5))) a isosceles right triangle | R = a/sqrt(2) isosceles triangle | R = 1/8 (a^2/h + 4 h) Kepler triangle | R = (a ϕ)/2 pennant | R = 1/8 (a^2/h + 4 h) right triangle | R = 1/2 sqrt(a^2 + b^2) scalene triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    30-60-90 triangle | a 3, 4, 5 triangle | 5 a equilateral triangle | a inverted equilateral triangle | a golden triangle | a ϕ isosceles right triangle | sqrt(2) a isosceles triangle | max(a, sqrt(a^2/4 + h^2)) Kepler triangle | a ϕ pennant | max(a, sqrt(a^2/4 + h^2)) right triangle | sqrt(a^2 + b^2) scalene triangle | max(a, b, c) triangle | max(a, b, c)

    30-60-90 triangle | χ = 1 3, 4, 5 triangle | χ = 1 equilateral triangle | χ = 1 inverted equilateral triangle | χ = 1 golden triangle | χ = 1 isosceles right triangle | χ = 1 isosceles triangle | χ = 1 Kepler triangle | χ = 1 pennant | χ = 1 right triangle | χ = 1 scalene triangle | χ = 1 triangle | χ = 1

    30-60-90 triangle | s^_ = (a (204 + 36 sqrt(3) + 81 log(3) + 2 (9 + 8 sqrt(3)) log(2 + sqrt(3))))/1440 3, 4, 5 triangle | s^_ = (a (20460 + 9728 log(2) + 5103 log(3)))/22500 equilateral triangle | s^_ = 1/20 a (3 + 3 log(3)) inverted equilateral triangle | s^_ = 1/20 a (3 + 3 log(3)) golden triangle | s^_ = 1/60 a ((5 + 2 sqrt(5)) log(1 + 2/sqrt(5)) + 2 (7 + sqrt(5) + sqrt(5) sinh^(-1)(2))) isosceles right triangle | s^_ = 1/30 a (2 + 4 sqrt(2) + (4 + sqrt(2)) sinh^(-1)(1)) isosceles triangle | s^_ = (-a^5 + 8 a^3 h^2 + 8 a^4 sqrt(a^2 + 4 h^2) + 2 h^2 ((a^2 + 4 h^2)^(3/2) log((a (sqrt(a^2 + 4 h^2) + a))/(2 h^2) + 1) + 16 a^3 log((sqrt(a^2 + 4 h^2) + a)/a)) + 48 a h^4)/(30 a (a^2 + 4 h^2)^(3/2)) Kepler triangle | s^_ = (8 a (7 sqrt(2) + 3 sqrt(10) + 25 sqrt(1 + sqrt(5)) + 11 sqrt(5 (1 + sqrt(5))) + sqrt(2) (7 + 3 sqrt(5)) log(1/2 (1 + sqrt(5) + sqrt(2 (1 + sqrt(5))))) + sqrt(11 + 5 sqrt(5)) log(9 + 4 sqrt(5) + 2 sqrt(38 + 17 sqrt(5))) + sqrt(199 + 89 sqrt(5)) sinh^(-1)(2)))/(15 (1 + sqrt(5))^(9/2)) right triangle | s^_ = (2 a^5 b + 4 a^3 b^3 + a b^4 (sqrt(a^2 + b^2) + 2 b) + a^4 b sqrt(a^2 + b^2) + (a^2 + b^2)^(3/2) (b^3 log((sqrt(a^2 + b^2) + a)/b) + a^3 log((sqrt(a^2 + b^2) + b)/a)) + 2 a^3 b^3 coth^(-1)((a + b)/sqrt(a^2 + b^2)))/(15 a b (a^2 + b^2)^(3/2)) scalene triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c) triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)

    3, 4, 5 triangle | A^_ = a^2/2 equilateral triangle | A^_ = a^2/(16 sqrt(3)) inverted equilateral triangle | A^_ = a^2/(16 sqrt(3)) golden triangle | A^_ = 1/48 sqrt(5 + 2 sqrt(5)) a^2 isosceles right triangle | A^_ = a^2/24 isosceles triangle | A^_ = (a h)/24 Kepler triangle | A^_ = (a^2 sqrt(ϕ))/24 pennant | A^_ = (a h)/24 right triangle | A^_ = (a b)/24 scalene triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

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