In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (t…
What is true about an equilateral triangle?
This is true for any equilateral triangle. Properties of equilateral triangles Equilateral triangles are regular polygons Since an equilateral triangle is also an equiangular triangle, it is a regular polygon. All equilateral triangles are similar Triangle ABC and triangle PQR are equiangular so, ABC ~ PQR. Altitudes of equilateral triangles
What are the identifying features of an equilateral triangle?
- The altitude, median, angle bisector, and perpendicular bisector for each side are all the same single line.
- These 3 lines (one for each side) are also the lines of symmetry of the triangle.
- All three of the lines mentioned above have the same length of s 3 2 \frac {s\sqrt {3}} {2} 2s 3 .
What are some facts about equilateral triangles?
Some interesting facts about equilateral triangles include: They are the only type of triangle that has equal angles that are all 60 degrees each. The triangle's sides are all of the same lengths, and always have a proportion of 1:1:1. This triangle has three lines of symmetry, meaning lines that divide it into perfect halves.
What is an equilateral triangle also called?
The word ‘Equilateral’ is formed by the combination of two words, i.e., “Equi” meaning equal and “Lateral” meaning sides. An equilateral triangle is also called a regular polygon or regular triangle since all its sides are equal. Suppose, ABC is an equilateral triangle, then, as per the definition;
What is meaning by equilateral?
having all sides equalDefinition of equilateral 1 : having all sides equal an equilateral triangle an equilateral polygon — see triangle illustration. 2 : having all the faces equal an equilateral polyhedron.
What is the meaning of isosceles triangle?
[count] mathematics. : a triangle in which two sides have the same length.
What are 2 equilateral triangles?
They both have one side equal in length to the larger equilateral triangle, and these sides are at an angle of 60^\text{o} to each other. They also both have one side equal in length to the smaller equilateral triangle, and these sides are also at an angle of 60^\text{o} to each other.
Where is the equilateral triangle?
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
What is the meaning of scalene triangle?
A scalene triangle is a triangle in which all three sides are in different lengths, and all three angles are of different measures. However, the sum of all the interior angles is always equal to 180 degrees.
What is the meaning of obtuse triangle?
An obtuse-angled triangle is a triangle in which one of the interior angles measures more than 90° degrees. In an obtuse triangle, if one angle measures more than 90°, then the sum of the remaining two angles is less than 90°.
What are the 3 types of triangle?
According to the lengths of their sides, triangles can be classified into three types which are: Scalene. Isosceles. Equilateral.
What are the 7 types of triangles?
Classification of Triangles Based on Side LengthsScalene Triangles. All sides of a scalene triangle are of different lengths. ... Isosceles Triangles. An isosceles triangle has two sides of equal length. ... Equilateral Triangles. ... Acute Triangles. ... Right Triangles. ... Obtuse Triangles.
What are the 4 types of triangle?
This math worksheet gives your child practice identifying equilateral, isosceles, scalene, and right triangles.
What is equilateral triangle Class 7?
The equilateral triangle is the triangle in which all sides have equal length; it is also equiangular i.e. angles are also equal. 4. If you are given all the angles equal.
What is perfect triangle?
In trigonometry, a perfect triangle is defined as a triangle that has a perimeter equal to its area. There are several ways of finding perfect triangles. A perfect triangle can be an equilateral, an isosceles, or a right triangle.
Is equilateral triangle a formula?
Formulas and Calculations for an Equilateral Triangle: Area of Equilateral Triangle Formula: K = (1/4) * √3 * a2. The altitude of Equilateral Triangle Formula: h = (1/2) * √3 * a. Angles of Equilateral Triangle: A = B = C = 60 degrees.
What is isosceles triangle class 9th?
The properties of an Isosceles Triangle are: The two sides of an Isosceles Triangle are congruent. Since the two sides are congruent, therefore, the two angles made by the two equal sides with the base are also congruent. The altitude of an Isosceles Triangle divides the triangle into two equal right-angled triangles.
What is the symbol of isosceles?
Isosceles triangleTypetriangleEdges and vertices3Schläfli symbol( ) ∨ { }Symmetry groupDih2, [ ], (*), order 22 more rows
What is an equilateral triangle?
A triangle that has all its sides equal in dimension and each angle measure up to 60 degrees, is called an equilateral triangle.
What is the area of equilateral triangle formula?
The formula to find area of equilateral triangle is given by: A = (√3/4)a2, where a is the length of side of equilateral triangle.
What is the perimeter of an equilateral triangle?
The perimeter of an equilateral triangle is the sum of all its three equal sides. Perimeter = 3a, where a is the length of its sides.
What is the height of equilateral triangle?
The formula to find the height of equilateral triangle is: Height, h = (√3/2)a, where a is the side of the equilateral triangle.
What type of polygon is an equilateral triangle?
An equilateral triangle is a regular polygon or a regular triangle.
What is the different name of triangles based on sides?
Based on sides, there are three different kinds of triangles. Their names are: Scalene triangle Isosceles triangle Equilateral triangle
How many sides does an equilateral triangle has?
All triangles have 3 sides only.
What is the perimeter of equilateral triangle with side equal to 10cm?
Perimeter = 3 x sides of equilateral triangle Perimeter = 3 x 10 = 30 cms
What do we mean by equilateral?
The term equilateral is made up of ‘equi,’ which means equal, and ‘lateral,’ which means side. So if a shape has all the sides of the same length i...
What are the properties of an equilateral triangle?
The three properties of an equilateral triangle are: All three sides are equal.All three angles are congruent.The figure has three lines of symmetry.
How do you know if a triangle is equilateral?
You can know that a triangle is equilateral by measuring its sides. If they are of equal lengths, then the triangle is equilateral.
Can a right triangle be equilateral?
An equilateral triangle can never be a right angled as a right-angled triangle has one of the angles as 90° whereas an equilateral triangle has all...
Is an equilateral triangle isosceles?
An equilateral triangle is a special case of an isosceles triangle having not just two, but all three sides and angles equal.
Let's learn!
In geometry, an equilateral triangle is a triangle whose all three sides are of equal length.
Let's do it!
Instead of handing out math geometry worksheets on triangles, ask children to observe/spot things around them in which they can see equilateral triangles.
What is the difference between an equilateral and an isosceles triangle?
In the scalene triangle, all the sides of the triangle are not equal and angles are also not equal. In the isosceles triangle, two sides are equal and the opposite angles of equal sides are equal.
How to find the area of an equilateral triangle?
The formula to find the area of an equilateral triangle is here:#N#Area of an equilateral triangle = √3a 2 / 4, where a is the side of of an equilateral triangle.
What is an equilateral triangle?
Equilateral Triangle. In geometry, a triangle with all equal sides said to be an equilateral triangle. If we break the word equilateral, here "equi" refers to equivalent and "lateral" refers sides. An equilateral triangle is also called a regular polygon with three sides equal.
Why is a polygon a regular polygon?
a regular polygon because it has three sides. the perpendicular drawn from any of the vertex to the opposite side of an equilateral triangle bisects the side in equal lengths. It also bisects the angle of the vertex into equal halves i.e. 30 degrees each from where the perpendicular is drawn.
Is an equilateral triangle a regular triangle?
Equilateral triangle is considered as a regular polygon or a regular triangle as angles are equal and sides are also equal. The triangles are categorized into three different types based on their sides. They are isosceles triangle, scalene triangle and equilateral triangle. Equilateral triangle is different from the isosceles and scalene triangle.
Is the Ortho-centre and the Centroid at the same point?
Ortho-centre and centroid are at the same point.
What is the meaning of the theorem of Pompeiu?
Pompeiu's theorem states that, if P is an arbitrary point in the plane of an equilateral triangle ABC but not on its circumcircle, then there exists a triangle with sides of lengths PA, PB, and PC. That is, PA, PB, and PC satisfy the triangle inequality that the sum of any two of them is greater than the third. If P is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem .
What is the only triangle with no Euler line connecting some of the centers?
Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular:
How to draw an equilateral triangle?
An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment
What is the internal angle of an equilateral triangle?
Internal angle ( degrees) 60°. In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
What is the ratio of the sum of the distances from the vertices to the sum of its distances?
Given a point P in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when P is the centroid. In no other triangle is there a point for which this ratio is as small as 2. This is the Erdős–Mordell inequality; a stronger variant of it is Barrow's inequality, which replaces the perpendicular distances to the sides with the distances from P to the points where the angle bisectors of ∠ APB, ∠ BPC, and ∠ CPA cross the sides ( A, B, and C being the vertices).
Which triangles have Steiner inellipses?
Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle).
Which triangle has the smallest ratio of the circumradius to the inradius?
By Euler's inequality, the equilateral triangle has the smallest ratio R / r of the circumradius to the inradius of any triangle: specifically, R / r = 2. : p.198
What is the area of a two dimensional triangle called?
The amount of space occupied in a two dimensional plane of an equilateral triangle is called the area of equiangular or equilateral triangle. The formula goes like this:
What is the equilateral triangle that makes 180 degrees?
Ans: Let ∆ABC is an equilateral triangle whose all the angles are equal i.e. <A=<B=<C that makes 180°.Your given to prove that it is an equilateral triangle.
What is a triangle that has all the sides and angles?
A triangle that’s all the sides and angles are equal and congruent to each other.
What are the three types of triangles?
Triangles are mainly classified into three types based on their sides. Scalene triangle, isosceles triangle and equilateral triangle. Here you are reading about equilateral triangles. But what makes equilateral triangles different from other triangles based on sides? The differentiation is:
What is the area of an equilateral triangle?
The area of an equilateral triangle is √3/4 (a)square.
What is the interior angle of a triangle?
A triangle that’s none of the sides are equal and congruent. Also the angles vary from each other. However, their interior angles measure 180°.
What is the opposite angle of the sides AB and AC?
Let, the side BC is not given to you. You have the sides AB and AC. So here you can conclude that the opposite angles of the sides AB and AC are equal. Therefore,<ABC=<ACB.
Different types of triangles
Before we talk about the meaning of triangles, we need to take a step back and remind ourselves of some basic geometry – since there are several different types of triangles, and the symbolism can vary, depending on which type appears.
Triangles and the number three
The main feature that makes triangles so important and meaningful is the fact that they have three points – the word “triangle” means “three angles” – and the number three is considered to be one of the most significant of all numbers.
Different meanings of the triangle Symbol
As we have seen, the triangle is an important shape loaded with symbolism and represents the number three, which is also one of the most significant numbers – so now let’s look at the symbolism of triangles in more depth.
A powerful symbol with many meanings
Triangles don’t appear very often in nature, but the power of this symbol has fascinated people since time immemorial.
What is the triangle in Buddhism?
The triangle also appears in Buddhism as Abhijñā which is access to direct knowledge through virtuous living and intensive meditation and concentration. Access to this appears as the Eye of Providence, which is an eye enclosed in a triangle that symbolizes Divine Vision.
Why are triangles associated with alchemy?
Triangles are often associated with alchemy because triangles are a heavily used symbol in differentiating various elements. In alchemy, the triangle represents the three realms of our existence that we encounter in our human experience. These are the body, the mind and the spirit/soul.
What is the shape of a triangle?
The shape of a triangle can be seen in ancient architecture, such as the pyramids of Egypt, which was believed to hold deep power in the spiritual realm . The triangle carries spiritual information regarding ascension to another realm after death, such as eternity or reincarnation.
What does the triangle represent?
A triangle represents manifestation, enlightenment, revelation, and a higher perspective. It is often used to mark the cycles of growth that lead to a higher state of being. Spiritually, it represents a path towards enlightenment or connection to an omnipresent being.
What is the purpose of the base of a triangle?
In another interpretation, the base of a triangle can be seen as a strong foundation that allows you to launch into a higher reality. In everyday life, triangles play a significant role in communicating important messages, such as: Recycle symbol to help you decide what can be reused instead of thrown away.
What is the meaning of triangles?
Triangles are one of the basic shapes that are seen all throughout our reality, especially in the realm of spirituality, religion, and symbolic imagery. It is seen so often that it is frequently asked what the triangle represents on a symbolic and spiritual level. Is there a deeper symbolic meaning to the triangle, ...
Why do we use triangles?
Triangles have been used as a symbol to communicate hidden messages for thousands of years. Symbols hold many different meanings depending on the context in which they are used to communicate. Depending on where you look, and the messages you need in order to move through life lessons, a triangle can have a different meaning for you.
Overview
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
Characterizations
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.
Notable theorems
Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.
Other properties
By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2.
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.
Geometric construction
An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment
In culture and society
Equilateral triangles have frequently appeared in man made constructions:
• The shape occurs in modern architecture such as the cross-section of the Gateway Arch.
• Its applications in flags and heraldry includes the flag of Nicaragua and the flag of the Philippines.
• It is a shape of a variety of road signs, including the yield sign.
See also
• Almost-equilateral Heronian triangle
• Isosceles triangle
• Ternary plot
• Trilinear coordinates