Equilateral Triangle Calculator

Height \( h \):

Area:

Perimeter:

Circumcircle Radius \( R \):

Incircle Radius \( r \):

1. What is an Equilateral Triangle Calculator?

Definition: This calculator computes the height, area, perimeter, circumcircle radius, and incircle radius of an equilateral triangle given its side length.

Purpose: It is used in geometry to determine the properties of an equilateral triangle, which is a triangle with all three sides equal.

2. How Does the Calculator Work?

The calculator uses the following formulas for an equilateral triangle with side \( a \):

Height \( h \): \( h = \frac{\sqrt{3}}{2} a \)
Area: \( \text{Area} = \frac{\sqrt{3}}{4} a^2 \)
Perimeter: \( \text{Perimeter} = 3a \)
Circumcircle Radius \( R \): \( R = \frac{a}{\sqrt{3}} \)
Incircle Radius \( r \): \( r = \frac{a \sqrt{3}}{6} \)

Unit Conversions:

Input Dimensions: m, cm (1 m = 100 cm), mm (1 m = 1000 mm), in (1 m = 39.3701 in), ft (1 m = 3.28084 ft), yd (1 m = 1.09361 yd)
Output Dimensions (Height, Perimeter, Radii): m, cm, mm, in, ft, yd
Output Area: m², cm² (1 m² = 10000 cm²), mm² (1 m² = 1000000 mm²), in² (1 m² = 1550.0031 in²), ft² (1 m² = 10.7639 ft²), yd² (1 m² = 1.19599 yd²)

Steps:

Input the length of side \( a \) with its unit.
Convert the dimension to meters for calculation.
Calculate height, area, perimeter, circumcircle radius, and incircle radius using the formulas.
Convert each result to its respective selected unit for display, using scientific notation for values less than 0.001, otherwise with 4 decimal places.

3. Importance of Equilateral Triangle Calculations

Calculating the properties of an equilateral triangle is crucial for:

Geometry Education: Understanding special triangles and their properties.
Engineering Design: Using equilateral shapes in structural components.
Art and Architecture: Creating symmetrical and aesthetically pleasing designs.

4. Using the Calculator

Examples:

Example 1: For a triangle with \( a = 8 \, \text{cm} \):
- Convert: \( a = 8 / 100 = 0.08 \, \text{m} \)
- Height \( h \): \( h = \frac{\sqrt{3}}{2} \times 0.08 \approx 0.0693 \, \text{m} \)
- Area: \( \text{Area} = \frac{\sqrt{3}}{4} \times 0.08^2 \approx 0.0028 \, \text{m}^2 \)
- Perimeter: \( \text{Perimeter} = 3 \times 0.08 = 0.24 \, \text{m} \)
- Circumcircle Radius \( R \): \( R = \frac{0.08}{\sqrt{3}} \approx 0.0462 \, \text{m} \)
- Incircle Radius \( r \): \( r = \frac{0.08 \sqrt{3}}{6} \approx 0.0231 \, \text{m} \)
- Convert: \( h = 6.9282 \, \text{cm} \), Area = \( 27.7128 \, \text{cm}^2 \), Perimeter = \( 24 \, \text{cm} \), \( R = 4.6188 \, \text{cm} \), \( r = 2.3094 \, \text{cm} \)
Example 2: For a triangle with \( a = 36 \, \text{in} \):
- Convert: \( a = 36 / 39.3701 = 0.9144 \, \text{m} \)
- Height \( h \): \( h = \frac{\sqrt{3}}{2} \times 0.9144 \approx 0.7919 \, \text{m} \)
- Area: \( \text{Area} = \frac{\sqrt{3}}{4} \times 0.9144^2 \approx 0.3623 \, \text{m}^2 \)
- Perimeter: \( \text{Perimeter} = 3 \times 0.9144 = 2.7432 \, \text{m} \)
- Circumcircle Radius \( R \): \( R = \frac{0.9144}{\sqrt{3}} \approx 0.5279 \, \text{m} \)
- Incircle Radius \( r \): \( r = \frac{0.9144 \sqrt{3}}{6} \approx 0.2640 \, \text{m} \)
- Convert: \( h = 31.1792 \, \text{in} \), Area = \( 561.2243 \, \text{in}^2 \), Perimeter = \( 108 \, \text{in} \), \( R = 20.7861 \, \text{in} \), \( r = 10.3931 \, \text{in} \)

5. Frequently Asked Questions (FAQ)

Q: What is an equilateral triangle?
A: An equilateral triangle is a triangle with all three sides equal and all three angles equal to 60°.

Q: Why are equilateral triangles important?
A: They have consistent side and angle ratios, making them useful in geometry, engineering, and design for simplifying calculations.