Equilateral Triangles: Meaning, Formula & Properties - StudySmarter

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Equilateral Triangles

There are many objects around us that are in the form of triangles like pizza slices, tower tops and roofs, and birthday banners. But sometimes we come across triangular shapes which look exactly the same in any direction we rotate it, like a nacho chip or traffic signs. Are these some kind of special form of triangles? Are they really equal in all ways? Let's find out.

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Are all angles in an equilateral triangle equal?

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Which of the following statement is correct?

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If a triangle is equiangular, does that mean it is also equilateral?

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All angles in an equilateral triangle are?

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Which of the following is the same as height in an equilateral triangle?

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 What is the perimeter of an equilateral triangle if the measure of one of its sides is 4cm? 

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Are all angles in an equilateral triangle equal?

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Which of the following statement is correct?

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• Cell Biology
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If a triangle is equiangular, does that mean it is also equilateral?

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All angles in an equilateral triangle are?

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Which of the following is the same as height in an equilateral triangle?

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 What is the perimeter of an equilateral triangle if the measure of one of its sides is 4cm? 

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Team Equilateral Triangles Teachers

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• Fact Checked Content
• Last Updated: 03.05.2022
• Published at: 05.05.2022
• 8 min reading time

• Applied Mathematics
• Calculus
• Decision Maths
• Discrete Mathematics
• Geometry
• 2 Dimensional Figures
• 3 Dimensional Vectors
• 3-Dimensional Figures
• Affine geometry
• Altitude
• Analytic geometry
• Angle Sum Property
• Angles in Circles
• Arc Measures
• Area and Volume
• Area of Circles
• Area of Circular Sector
• Area of Parallelograms
• Area of Plane Figures
• Area of Rectangles
• Area of Regular Polygons
• Area of Rhombus
• Area of Trapezoid
• Area of a Kite
• Birational geometry
• Central Angle
• Complex geometry
• Composition
• Computational geometry
• Cones
• Conformal geometry
• Congruence Transformations
• Congruent Triangles
• Convex geometry
• Convexity in Polygons
• Coordinate Systems
• Cube Properties
• Cylinders
• Differential geometry
• Differential topology
• Dilations
• Discrete geometry
• Distance and Midpoints
• Equation Of Circles
• Equilateral Triangles
• Ergodic theory
• Euclidean geometry
• Exterior Angle Theorem
• Figures
• Finsler geometry
• Fractal geometry
• Fundamentals of Geometry
• Galois geometry
• Geodesy
• Geometric Inequalities
• Geometric Mean
• Geometric Probability
• Geometric Transformations
• Geometric analysis
• Geometric function theory
• Geometric topology
• Glide Reflections
• HL ASA and AAS
• Harmonic analysis
• Hodge theory
• Hyperbolic geometry
• Identity Map
• Inscribed Angles
• Integral geometry
• Interior Angles
• Inversive geometry
• Isometry
• Isosceles Triangles
• Kähler manifolds
• Law of Cosines
• Law of Sines
• Linear Measure and Precision
• Manifolds
• Mathematical Physics
• Median
• Metric geometry
• Minimal surfaces
• Mirror symmetry
• Moduli spaces
• Morse theory
• Möbius transformations
• Non-Euclidean geometry
• Parallel Lines Theorem
• Parallelograms
• Perpendicular Bisector
• Plane Geometry
• Poisson geometry
• Polygon Properties
• Polygons
• Projections
• Projective geometry
• Properties of Chords
• Proportionality Theorems
• Pyramids
• Pythagoras Theorem
• Quantum geometry
• Quaternionic analysis
• Rectangle
• Rectangular Prism
• Reflection in Geometry
• Regular Polygon
• Rhombus Properties
• Rhombuses
• Ricci flow
• Right Triangles
• Rotations
• SSS and SAS
• Sasaki geometry
• Scalene Triangles
• Segment Length
• Similarity
• Similarity Transformations
• Simplicial complexes
• Special Right Triangles
• Special quadrilaterals
• Spherical geometry
• Square Properties
• Squares
• Sub-Riemannian geometry
• Surface Area of Cone
• Surface Area of Cylinder
• Surface Area of Prism
• Surface Area of Sphere
• Surface Area of a Solid
• Surface of Pyramids
• Symmetry
• Symplectic geometry
• Toric geometry
• Transformation groups
• Translations
• Transversal Lines
• Trapezoid Properties
• Trapezoids
• Triangle Inequalities
• Triangles
• Twistor theory
• Using Similar Polygons
• Vector Addition
• Vector Product
• Vector bundles
• Volume of Cone
• Volume of Cylinder
• Volume of Pyramid
• Volume of Solid
• Volume of Sphere
• Volume of prisms
• Vortex dynamics
• Weyl geometry
• What is Point Slope Form
• Logic and Functions
• Mechanics Maths
• Probability and Statistics
• Pure Maths
• Statistics
• Theoretical and Mathematical Physics

Contents

• Applied Mathematics
• Calculus
• Decision Maths
• Discrete Mathematics
• Geometry
• 2 Dimensional Figures
• 3 Dimensional Vectors
• 3-Dimensional Figures
• Affine geometry
• Altitude
• Analytic geometry
• Angle Sum Property
• Angles in Circles
• Arc Measures
• Area and Volume
• Area of Circles
• Area of Circular Sector
• Area of Parallelograms
• Area of Plane Figures
• Area of Rectangles
• Area of Regular Polygons
• Area of Rhombus
• Area of Trapezoid
• Area of a Kite
• Birational geometry
• Central Angle
• Complex geometry
• Composition
• Computational geometry
• Cones
• Conformal geometry
• Congruence Transformations
• Congruent Triangles
• Convex geometry
• Convexity in Polygons
• Coordinate Systems
• Cube Properties
• Cylinders
• Differential geometry
• Differential topology
• Dilations
• Discrete geometry
• Distance and Midpoints
• Equation Of Circles
• Equilateral Triangles
• Ergodic theory
• Euclidean geometry
• Exterior Angle Theorem
• Figures
• Finsler geometry
• Fractal geometry
• Fundamentals of Geometry
• Galois geometry
• Geodesy
• Geometric Inequalities
• Geometric Mean
• Geometric Probability
• Geometric Transformations
• Geometric analysis
• Geometric function theory
• Geometric topology
• Glide Reflections
• HL ASA and AAS
• Harmonic analysis
• Hodge theory
• Hyperbolic geometry
• Identity Map
• Inscribed Angles
• Integral geometry
• Interior Angles
• Inversive geometry
• Isometry
• Isosceles Triangles
• Kähler manifolds
• Law of Cosines
• Law of Sines
• Linear Measure and Precision
• Manifolds
• Mathematical Physics
• Median
• Metric geometry
• Minimal surfaces
• Mirror symmetry
• Moduli spaces
• Morse theory
• Möbius transformations
• Non-Euclidean geometry
• Parallel Lines Theorem
• Parallelograms
• Perpendicular Bisector
• Plane Geometry
• Poisson geometry
• Polygon Properties
• Polygons
• Projections
• Projective geometry
• Properties of Chords
• Proportionality Theorems
• Pyramids
• Pythagoras Theorem
• Quantum geometry
• Quaternionic analysis
• Rectangle
• Rectangular Prism
• Reflection in Geometry
• Regular Polygon
• Rhombus Properties
• Rhombuses
• Ricci flow
• Right Triangles
• Rotations
• SSS and SAS
• Sasaki geometry
• Scalene Triangles
• Segment Length
• Similarity
• Similarity Transformations
• Simplicial complexes
• Special Right Triangles
• Special quadrilaterals
• Spherical geometry
• Square Properties
• Squares
• Sub-Riemannian geometry
• Surface Area of Cone
• Surface Area of Cylinder
• Surface Area of Prism
• Surface Area of Sphere
• Surface Area of a Solid
• Surface of Pyramids
• Symmetry
• Symplectic geometry
• Toric geometry
• Transformation groups
• Translations
• Transversal Lines
• Trapezoid Properties
• Trapezoids
• Triangle Inequalities
• Triangles
• Twistor theory
• Using Similar Polygons
• Vector Addition
• Vector Product
• Vector bundles
• Volume of Cone
• Volume of Cylinder
• Volume of Pyramid
• Volume of Solid
• Volume of Sphere
• Volume of prisms
• Vortex dynamics
• Weyl geometry
• What is Point Slope Form
• Logic and Functions
• Mechanics Maths
• Probability and Statistics
• Pure Maths
• Statistics
• Theoretical and Mathematical Physics

Contents

• Fact Checked Content
• Last Updated: 03.05.2022
• 8 min reading time

• Content creation process designed by Lily Hulatt
• Content sources cross-checked by Gabriel Freitas
• Content quality checked by Gabriel Freitas

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Jump to a key chapter

In geometry, triangles can be classified into different forms based on their sides and angles. And one of these forms is an equilateral triangle. In this section, we will understand the concept of the equilateral triangle and see its properties and formulas based on it.

A triangle is an equilateral triangle if it has three congruent sides. In other words, if all three sides of a triangle are of the same length, then it is an equilateral triangle.

So the name equilateral is derived from equi, which means equal, and lateral, means sides.

Equilateral triangle with congruent sides, Mouli Javia - StudySmarter Originals

Equilateral triangles and angles

We can classify equilateral triangles based on their angles too.

An equilateral triangle is a triangle with all three of its internal angles congruent and equal to60° .

Equilateral Triangle with same angles, Mouli Javia - StudySmarter Originals

Corollaries on equilateral triangles

Let's take a look at some important statements regarding equilateral triangles

Corollary 1

Statement: Each angle of an equilateral triangle is60°.

Proof: To prove this consider∆XYZan equilateral triangle.

∴ XY=YZ=ZX

Now, an equilateral triangle is also considered an isosceles triangle. So we can apply the properties of the isosceles triangle to the equilateral triangle. Here we use Isosceles Triangle Theorem.

For that take:

XY=YZ and YZ=ZX

∴ ∠Z=∠X and ∠X=∠Y

⇒ ∠X=∠Y=∠Z

Now consider one of the properties of a triangle, which states that the sum of all internal angles of a triangle is equal to180°:∠X+∠Y+∠Z=180°

As all three angles are equal, we can just consider one of them instead of all.

∴∠X+∠X+∠X=180°

⇒ 3∠X=180°

⇒ ∠X=180°3

∴ ∠X=60°

So, ∠X=∠Y=∠Z=60°.

Hence, we can say an equilateral triangle is an equiangular triangle.

The Isosceles Triangle theorem states that angles opposite to the two sides of a triangle are equal if these two sides are equal.

From this corollary, we arrive at the next corollary.

Corollary 2

Statement: A triangle is equiangular if and only if it is equilateral.

Equilateral Triangle Properties

Here are some of the properties of equilateral triangles:

1. An equilateral triangle is a regular polygon as it has three sides.

2. All sides and angles of equilateral triangles are congruent.

3. A perpendicular line drawn from any vertex of an equilateral triangle to its opposite side bisects both side and angle.

4. This perpendicular line (as mentioned above) is the same line for altitude, median, perpendicular bisector, and angle bisector for the same side.

5. Lines of symmetry in equilateral triangles are the three mentioned lines from each side.

6. In equilateral triangles, the centroid, ortho-center, circumcenter, and incenter are at the same point.

Remember that to bisect means to divide or split into two equal parts.

Equilateral Triangle Formulas

Let's discuss a few formulas related to equilateral triangles, including its:

• Perimeter
• Area
• Height

Perimeter of an Equilateral Triangle

Perimeter is the sum of all the sides. And as we are talking about an equilateral triangle, here all sides are equal. So the perimeter of an equilateral triangle is three times the length of one side.

Perimeter of an equilateral triangle=3a. Herea is the side length.

From this, we can deduce the formula for Semi Perimeter. Semi Perimeter is half of the perimeter of an equilateral triangle and we can calculate it as follows.

Semi Perimeter of an equilateral triangle= 3a2

We usually use the semi perimeter to calculate the area of a triangle using Heron's formula.

What is the perimeter for the given equilateral triangle with a side of 6 cm? Also, find the semi perimeter for it.

Equilateral triangle, Mouli Javia - StudySmarter Originals

Solution: Herea=6cm. So applying the formula of perimeter, we get:

Perimeter of an equilateral triangleid="2552506" role="math" = 3a = 3×6 = 18 cm.

Semi perimeter of an equilateral triangleid="2552508" role="math" alt="" =3a2 = 182 =9 cm.

Area of an Equilateral Triangle

Area is calculated to measure the space occupied within the sides of a polygon in a 2D plane. The formula to find the area of an equilateral triangle is as follows.

Area of an equilateral triangle= 34a2, wherea is side length.

We can also calculate area using Heron's formula if a semi perimeter is given. Heron's formula is as follows.

Area of an equilateral triangle=s(s-a)3, where a is the side length and s is the semi perimeter of the triangle.

Calculate the area for an equilateral triangle with a side of 5 cm.

Equilateral triangle, Mouli Javia - StudySmarter Originals

Solution: Herea=5 cm.

Area of an equilateral triangle = 34a2

= 3452

= 10.83

Therefore, the area of a given equilateral triangle is10.83 cm2.

Height of an Equilateral Triangle

The height of an equilateral triangle is the perpendicular distance from a vertex of that triangle to its opposite side.

Height of equilateral triangle, Mouli Javia - StudySmarter Originals

The formula to calculate the height of an equilateral triangle is given below.

Height of an equilateral triangle= 32a, wherea is the side length.

Find the height of an equilateral triangle with a side length of 15 cm.

Solution: Using the formula of height, we can say:

Height of an equilateral triangle= 32a

= 3214

= 73 = 12.12 cm

Hence the height (or altitude) of an equilateral triangle is 12.12 cm.

Examples of equilateral triangles

Now we work on some examples based on the above theory.

Find the area of an equilateral triangle that has a perimeter of 18 cm.

Solution: To find the area of an equilateral triangle, we need to know the length of its sides. So first we will find the side length using perimeter. We know that the formula for the perimeter of an equilateral triangle is3a. And the value of perimeter is also given in the question, which is 18 cm.

∴ 18=3a

⇒ a=183⇒ a=6 cm

Now as we have found the side length, we can use it in the formula of the area to calculate it.

Area of an equilateral triangle = 34a2

= 3462

= 93 = 15.58 cm2

Hence, an equilateral triangle that has a perimeter of 18 cm, has an area of 15.58 cm2.

An equilateral triangle with two side lengths is given. The length of one side is(3x+8) and for the other side is(4x+7). What is the measure of side length for this equilateral triangle? Also, find the perimeter for this triangle.

Solution: As the given triangle is an equilateral triangle, we know that all the sides of it are equal. So the given two side lengths are equal, and the equations can be set as equal to one another as well.

⇒ 3x+8=4x+7

To determine the side length we solve the above equation and find the value of x.

⇒ 4x-3x=8-7⇒x=1

Now, as both the side lengths are equal, we substitute the value of x in any one of the side lengths.

By substituting in3x+8, we get

3x+8 = 31+8 = 11.

We can check the correctness of the found value of x, by substituting x in both the side lengths. If both the value of side lengths are equal, the value of x is correct. Let's see for our case. We have already found the value of one of the side lengths. Let's find the other side length and compare it.

Substituting x in 4x+7, we again get the value of 11. Hence as both the values of side length are equal, our calculated x value is correct!

Now that we have the sides' length, we can easily calculate the perimeter of the equilateral triangle.

Perimeter of an equilateral triangle= 3a. Herea=11.

⇒ 3a = 3(11) = 33.

So, the perimeter of the given equilateral triangle is 33 cm.

Equilateral triangles - Key takeaways

• A triangle is an equilateral triangle if it has three congruent sides.
• An equilateral triangle is a triangle with all three of its internal angles congruent and equal to60° .
• A triangle is equiangular if and only if it is equilateral.
• The perimeter of an equilateral triangle is3a.
• The semi perimeter of an equilateral triangle is 3a2.
• The area of an equilateral triangle is34a2.
• The area of an equilateral triangle (using Heron's formula) is ss-a3.
• The height of an equilateral triangle is32a.

Similar topics in Math

• Probability and Statistics
• Statistics
• Mechanics Maths
• Geometry
• Calculus
• Pure Maths
• Decision Maths
• Logic and Functions
• Discrete Mathematics
• Theoretical and Mathematical Physics
• Applied Mathematics

Related topics to Geometry

• Glide Reflections
• Symmetry
• Area of a Kite
• Polygons
• Altitude
• Triangles
• Trapezoids
• Area of Regular Polygons
• Surface Area of a Solid
• Area of Parallelograms
• Isometry
• Congruence Transformations
• Volume of Solid
• Volume of Sphere
• Regular Polygon
• Equilateral Triangles
• Rotations
• Area of Plane Figures
• Surface Area of Sphere
• Rectangle
• Distance and Midpoints
• Identity Map
• Translations
• Area of Trapezoid
• SSS and SAS
• Volume of Cylinder
• Vector Product
• Properties of Chords
• Plane Geometry
• Arc Measures
• Dilations
• Composition
• Volume of prisms
• Right Triangles
• Volume of Cone
• Congruent Triangles
• 2 Dimensional Figures
• Convexity in Polygons
• Fundamentals of Geometry
• Area and Volume
• Median
• Perpendicular Bisector
• Geometric Probability
• Area of Circular Sector
• Area of Rhombus
• Law of Sines
• Linear Measure and Precision
• Pythagoras Theorem
• Parallelograms
• Law of Cosines
• Geometric Mean
• Isosceles Triangles
• Parallel Lines Theorem
• Special quadrilaterals
• Proportionality Theorems
• Similarity Transformations
• Volume of Pyramid
• Squares
• 3-Dimensional Figures
• Rhombuses
• 3 Dimensional Vectors
• Surface Area of Prism
• Inscribed Angles
• Coordinate Systems
• Surface Area of Cone
• Segment Length
• Projections
• Similarity
• HL ASA and AAS
• Using Similar Polygons
• Equation Of Circles
• Angles in Circles
• Vector Addition
• Surface Area of Cylinder
• Reflection in Geometry
• Surface of Pyramids
• Figures
• Triangle Inequalities
• Area of Rectangles
• Area of Circles
• What is Point Slope Form
• Geometric Inequalities
• Computational geometry
• Sasaki geometry
• Kähler manifolds
• Integral geometry
• Mirror symmetry
• Quaternionic analysis
• Geodesy
• Manifolds
• Non-Euclidean geometry
• Discrete geometry
• Geometric topology
• Conformal geometry
• Symplectic geometry
• Projective geometry
• Inversive geometry
• Ricci flow
• Euclidean geometry
• Quantum geometry
• Finsler geometry
• Hyperbolic geometry
• Harmonic analysis
• Toric geometry
• Simplicial complexes
• Möbius transformations
• Minimal surfaces
• Fractal geometry
• Metric geometry
• Galois geometry
• Differential geometry
• Convex geometry
• Twistor theory
• Vortex dynamics
• Complex geometry
• Weyl geometry
• Transformation groups
• Differential topology
• Ergodic theory
• Affine geometry
• Geometric function theory
• Hodge theory
• Poisson geometry
• Mathematical Physics
• Geometric analysis
• Spherical geometry
• Vector bundles
• Birational geometry
• Analytic geometry
• Sub-Riemannian geometry
• Moduli spaces
• Morse theory
• Cones
• Scalene Triangles
• Geometric Transformations
• Trapezoid Properties
• Square Properties
• Cylinders
• Rectangular Prism
• Cube Properties
• Central Angle
• Transversal Lines
• Rhombus Properties
• Polygon Properties
• Pyramids
• Angle Sum Property
• Interior Angles
• Exterior Angle Theorem
• Special Right Triangles

Flashcards in Equilateral Triangles

6 Start learning

Are all angles in an equilateral triangle equal?

No

Which of the following statement is correct?

Equilateral triangle is a isosceles triangle

If a triangle is equiangular, does that mean it is also equilateral?

No

All angles in an equilateral triangle are?

Acute angles

Which of the following is the same as height in an equilateral triangle?

Altitude

 What is the perimeter of an equilateral triangle if the measure of one of its sides is 4cm? 

12cm

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Frequently Asked Questions about Equilateral Triangles

What is an equilateral triangle?

An equilateral triangle is a triangle with equal sides and angles.

How to find the area of an equilateral triangle?

The area of an equilateral triangle is calculated by the following formulas:

Area = (√3/4) × (side)2

How many lines of symmetry does an equilateral triangle have?

An equilateral triangle has three lines of symmetry.

What is the rule for equilateral triangles?

In an equilateral triangle, all three sides are congruent. Also, all angles are equal and have a measure of 60°.

What is the formula for equilateral triangles?

The formulas for an equilateral triangle are as follows:

Perimeter = 3 × side

Area = (√3/4) × (side)2

Height = (√3/2) × side

Save Article Test your knowledge with multiple choice flashcards

Are all angles in an equilateral triangle equal?

A. No B. Yes

Which of the following statement is correct?

A. Isosceles triangle is an equilateral triangle B. Isosceles triangle is a special case of equilateral triangle C. Equilateral triangle is a special case of isosceles triangle D. Equilateral triangle is a isosceles triangle

If a triangle is equiangular, does that mean it is also equilateral?

A. Yes B. No

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• Checked by StudySmarter Editorial Team

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