Properties of Equilateral Triangle
An equilateral triangle has all three sides of equal length and all three angles of equal measure. Since the angle sum of any triangle is 180°, each angle of an equilateral triangle is 180° ÷ 3 = 60°.
It is the most symmetric triangle — it has 3 lines of symmetry and looks the same when rotated by 120° or 240°.
Equilateral triangles appear everywhere in nature and design — from honeycomb cells to traffic signs to the faces of a tetrahedron.
What is Properties of Equilateral Triangle - Grade 7 Maths (Triangles)?
Definition: An equilateral triangle is a triangle in which all three sides are equal and all three angles are 60°.
Properties:
- All sides are equal: AB = BC = CA.
- All angles are equal: ∠A = ∠B = ∠C = 60°.
- It has 3 lines of symmetry.
- It has rotational symmetry of order 3.
- The altitude, median, angle bisector, and perpendicular bisector from each vertex all coincide.
- The centroid, orthocentre, circumcentre, and incentre all lie at the same point.
Properties of Equilateral Triangle Formula
Formulas:
Perimeter = 3a
Area = (√3/4) × a²
Height = (√3/2) × a
Where a = length of each side.
Types and Properties
Relationship to Other Triangles:
- An equilateral triangle is a special case of an isosceles triangle (any two sides are equal).
- It is also a regular polygon (specifically, a regular 3-sided polygon).
- It is always an acute triangle (all angles < 90°).
Solved Examples
Example 1: Finding Perimeter
Problem: An equilateral triangle has side 7 cm. Find its perimeter.
Solution:
- Perimeter = 3 × 7 = 21 cm
Answer: Perimeter = 21 cm.
Example 2: Finding Area
Problem: Find the area of an equilateral triangle with side 6 cm.
Solution:
- Area = (√3/4) × 6² = (√3/4) × 36 = 9√3
- 9 × 1.732 = 15.588 ≈ 15.59 cm²
Answer: Area ≈ 15.59 cm² (or 9√3 cm²).
Example 3: Finding Height
Problem: Find the height of an equilateral triangle with side 10 cm.
Solution:
- Height = (√3/2) × 10 = 5√3
- 5 × 1.732 = 8.66 cm
Answer: Height = 8.66 cm (or 5√3 cm).
Example 4: Finding Side from Perimeter
Problem: The perimeter of an equilateral triangle is 36 cm. Find each side.
Solution:
- Side = Perimeter ÷ 3 = 36 ÷ 3 = 12 cm
Answer: Each side = 12 cm.
Example 5: Angle Verification
Problem: Verify that the angles of an equilateral triangle are each 60°.
Solution:
- All sides are equal → all angles are equal (isosceles triangle theorem applied twice).
- Let each angle = x. Then: x + x + x = 180°
- 3x = 180°, x = 60°
Answer: Each angle = 60°. Verified.
Example 6: Word Problem
Problem: A triangular garden has all sides equal to 15 m. Find the area of the garden.
Solution:
- Area = (√3/4) × 15² = (√3/4) × 225 = 56.25√3
- ≈ 56.25 × 1.732 ≈ 97.43 m²
Answer: Area ≈ 97.43 m².
Real-World Applications
Real-world uses:
- Honeycomb: Bees build hexagonal cells made of equilateral triangles — strongest shape per unit material.
- Traffic signs: Yield and warning signs are equilateral triangles.
- Architecture: Geodesic domes are built using equilateral triangles.
- Music: The triangle instrument is shaped as an equilateral triangle.
Key Points to Remember
- All sides equal, all angles = 60°.
- Perimeter = 3a, Area = (√3/4)a², Height = (√3/2)a.
- 3 lines of symmetry and rotational symmetry of order 3.
- Every equilateral triangle is also isosceles and acute.
- Centroid, orthocentre, circumcentre, and incentre all coincide.
- It is a regular polygon (regular 3-gon).
Practice Problems
- Find the perimeter of an equilateral triangle with side 9 cm.
- Find the area of an equilateral triangle with side 8 cm.
- The height of an equilateral triangle is 5√3 cm. Find its side.
- The perimeter of an equilateral triangle is 45 cm. Find its area.
- How many lines of symmetry does an equilateral triangle have?
Frequently Asked Questions
Q1. What is an equilateral triangle?
A triangle with all three sides equal and all three angles equal to 60°.
Q2. Is every equilateral triangle isosceles?
Yes. Since all three sides are equal, any pair of sides is equal, satisfying the isosceles definition.
Q3. What is the formula for area of an equilateral triangle?
Area = (√3/4) × a², where a is the side length.
Q4. Can an equilateral triangle be right-angled?
No. All angles are 60°. No angle is 90°.
Q5. How many lines of symmetry does it have?
3 lines of symmetry — each from a vertex to the midpoint of the opposite side.
Related Topics
- Properties of Isosceles Triangle
- Classification of Triangles
- Angle Sum Property of Triangle
- Area of Equilateral Triangle
- Exterior Angle Property of Triangle
- Triangle Inequality Property
- Medians and Altitudes of Triangle
- Right-Angled Triangle Property
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
- Basic Proportionality Theorem (BPT)
- Converse of Basic Proportionality Theorem
- Criteria for Similarity of Triangles
