In geometry, we come across many different shapes when we deal with figures. Some of these shapes have equal side lengths while some have equal angles; some of them are made up of straight lines while some have curves. In this page, we will understand the meaning of equilateral by using different polygons that have equal side lengths.
|1.||Definition of Equilateral|
|3.||Solved Examples on Equilateral|
|4.||Practice Questions on Equilateral|
|5.||FAQs on Equilateral|
Definition of Equilateral
The word equilateral is made up of two words, "equi" which means equal and "lateral" which means sides. Equilateral is defined as any shape which has all its sides equal.
We know that polygons are shapes that are made up of straight line segments, and a triangle is a polygon with the minimum number of sides. Now, if the side lengths of the polygons are equal, we call them equilateral shapes. Let's explore these equilateral shapes and their properties in detail now.
An equilateral triangle is a regular polygon with 3 equal sides.
The figure given above shows that triangle PQR is an equilateral triangle. Hence, the equal sides can be written as: PQ = QR= RP.
It should be noted that all the angles of this triangle are also equal. ∠PQR = ∠QRP = ∠RPQ = 60°
An equilateral quadrilateral is a polygon with 4 equal sides and it is called a rhombus. The figure given below shows that rhombus PQRS is an equilateral quadrilateral.
We cannot say that all the angles of an equilateral quadrilateral are equal. A square is a special case of equilateral quadrilateral in which all the four angles are equal. Thus, it can be said that if a quadrilateral is equilateral and equiangular, it is a square.
Square ABCD is an equilateral quadrilateral with all the four angles equal to 90°.
An equilateral pentagon is a polygon with 5 equal sides. The following figure shows that pentagon ABCDE is an equilateral pentagon.
The interior angles of an equilateral pentagon may or may not be equal. The special case of an equilateral pentagon in which all the five angles are also equal is called a regular pentagon.
Topics Related to Equilateral
Tips to Remember
- All the angles of an equilateral triangle are equal to 60°.
- Equiangular shapes are shapes in which all the interior angles are equal.
- A square is a special case of an equilateral quadrilateral in which all the angles and sides are equal.
- If a rhombus is equiangular, it is a square.
Example 1: Tim draws an equilateral triangle and measures one of the sides of the triangle to be 10 units. If the other two sides are of lengths (2x - 6) and (3y - 2), find the values of x and y.
Triangle ABC is an equilateral triangle, hence AB = BC = CA.
2x - 6 = 10 = 3y - 2
Let's solve each of these equations separately
2x - 6 = 10
2x = 10 + 6
2x = 16
x = 8
3y - 2 = 10
3y = 10 + 2
3y = 12
y = 4
\(\therefore\) The values of x and y are, 8 and 4 respectively.
Example 2: Paul is fencing his backyard with an iron fence. The total length of the iron fence required is 100 yards. If the backyard is in the shape of an equilateral quadrilateral, find the length of one side of the backyard.
Solution: Since the backyard is in the shape of an equilateral quadrilateral, all sides of the backyard are equal. Let us assume one of the sides of the backyard to be x.
The perimeter of the backyard will be = 4x
It is given that the total length of the iron fence is 100 yards.
Hence, 4x = 100
x = 25 yards
\(\therefore\) The length of one side of backyard is 25 yards.
Example 3: Find the area of an equilateral triangle with one side equal to 4 units.
Solution: It is given that the side of the equilateral triangle is 4 units.
Area of an equilateral triangle with side a is (√3/4)×a2
Putting the value of a = 4 in the equation.
Area = (√3/4)×a2 = (√3/4)× 42 = √3 × 4
Area = 4√3 unit2
Frequently Asked Questions (FAQs)
What is an Equilateral Parallelogram Called?
An equilateral parallelogram is known as a rhombus. A square is a special case of an equilateral rhombus in which all the angles are equal to 90°.
Is an Equilateral Triangle Also a Rhombus?
No, an equilateral triangle is not a rhombus. An equilateral triangle is a 3-sided polygon while a rhombus is a 4-sided polygon.
Are All Regular Polygons Equilateral?
Yes, all regular polygons are equilateral. A regular triangle is called an equilateral triangle, and a regular quadrilateral is called a square.
Is a Square an Equilateral?
Yes, a square is an equilateral quadrilateral with all four sides equal and all four angles equal.
What is the Difference Between Equilateral and Equiangular?
Equilateral figures are the figures with equal side lengths. Equiangular figures are the figures with equal angles.
Do Equilateral Triangles have Equal Angles?
Yes, an equilateral triangle has equal angles. Each angle of an equilateral triangle is 60°.
Are Right Triangles Equilateral?
No, right triangles are not equilateral. In a right-angled triangle, the length of the hypotenuse is always greater than the other two sides whereas in an equilateral triangle all the sides are equal. So, a right-angled triangle does not follow the rules for an equilateral triangle.