17. Equilateral Triangle Formulas
An equilateral triangle is a triangle in which all three sides are equal.
If each side of the equilateral triangle is represented by a, then:
Each interior angle of an equilateral triangle is 60°.
1. Equilateral Triangle Diagram
2. Basic Properties
| Property | Value |
|---|---|
| Number of sides | 3 |
| Equal sides | a = a = a |
| Each angle | 60° |
| Sum of angles | 180° |
| Height | √3a / 2 |
3. Perimeter of an Equilateral Triangle
Since all three sides are equal:
Where a is the length of one side.
Example 1: Find the Perimeter
The side of an equilateral triangle is 12 cm. Find its perimeter.
Given:
Formula:
Substitution:
Answer: 36 cm
4. Area of an Equilateral Triangle
The area of an equilateral triangle is:
Here a represents the side of the triangle.
This formula can also be obtained from:
For an equilateral triangle:
Therefore:
Example 2: Find the Area
Find the area of an equilateral triangle whose side is 8 cm.
Formula:
Substitute a = 8:
Answer: 16√3 cm²
5. Height of an Equilateral Triangle
The perpendicular height of an equilateral triangle is:
The height divides the equilateral triangle into two right-angled triangles.
Therefore, the base is divided into two equal parts:
Example 3: Find the Height
Find the height of an equilateral triangle whose side is 10 cm.
Answer: 5√3 cm
6. Inradius of an Equilateral Triangle
The inradius is the radius of the circle inscribed inside the triangle.
Since:
The inradius can also be written as:
Example 4: Find the Inradius
Find the inradius of an equilateral triangle whose side is 6 cm.
Answer: √3 cm
7. Circumradius of an Equilateral Triangle
The circumradius is the radius of the circle passing through all three vertices of the triangle.
It can also be expressed in terms of height:
Example 5: Find the Circumradius
Find the circumradius of an equilateral triangle whose side is 9 cm.
Answer: 3√3 cm
8. Important Relations
9. Equilateral Triangle Formula Table
| Quantity | Formula |
|---|---|
| Area | √3a² / 4 |
| Perimeter | 3a |
| Height | √3a / 2 |
| Inradius | a / (2√3) = h/3 |
| Circumradius | a / √3 = 2h/3 |
10. Exam Shortcuts
Shortcut 1 — Side to Height
If the side is given, you can directly calculate the height.
Shortcut 2 — Side to Area
No need to calculate height separately.
Shortcut 3 — Side to Inradius
Shortcut 4 — Side to Circumradius
11. More Solved Examples
Example 6
The perimeter of an equilateral triangle is 45 cm. Find the side.
Answer: 15 cm
Example 7
The side of an equilateral triangle is 14 cm. Find its height.
Answer: 7√3 cm
Example 8
Find the area of an equilateral triangle whose side is 4 cm.
Answer: 4√3 cm²
12. Practice Questions
1. Find the perimeter of an equilateral triangle whose side is 18 cm.
2. Find the area of an equilateral triangle whose side is 6 cm.
3. Find the height of an equilateral triangle whose side is 12 cm.
4. Find the inradius of an equilateral triangle whose side is 8 cm.
5. Find the circumradius of an equilateral triangle whose side is 6 cm.
6. The perimeter of an equilateral triangle is 36 cm. Find its side.
7. The side of an equilateral triangle is 10 cm. Find its area.
13. Answers
| Question | Answer |
|---|---|
| 1 | 54 cm |
| 2 | 9√3 cm² |
| 3 | 6√3 cm |
| 4 | 4√3/3 cm |
| 5 | 2√3 cm |
| 6 | 12 cm |
| 7 | 25√3 cm² |
14. Common Mistakes to Avoid
1. Do not confuse the side with the height.
2. For area, remember the factor √3/4.
3. Perimeter is 3a, not a².
4. Inradius and circumradius are different.
5. Area is expressed in square units such as cm².
6. Height is expressed in linear units such as cm.
15. Quick Revision
Source: The formulas in this section are based on the Equilateral Triangle section of the provided Mensuration Formula PDF. The source specifically lists formulas for area, perimeter, height, inradius and circumradius.
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