Introduction
In geometry, the circumradius of a triangle is the radius of the circle that passes through all three vertices of the triangle. This circle is called the circumcircle, and its center is known as the circumcenter. The circumradius plays an important role in trigonometry, coordinate geometry, and real-life applications such as navigation and engineering. In this blog, we’ll explore how to calculate the circumradius of different triangles with formulas and examples.
Circumradius Formula for Any Triangle
The general formula for the circumradius (R) of a triangle with sides a, b, and c, and area Δ is:
\(\displaystyle R = \frac{abc}{4\Delta}\)
Where:
→ a, b, c = sides of the triangle
→ Δ = area of the triangle (using Heron’s formula if needed)
This formula works for any triangle – scalene, isosceles, or equilateral.
Circumradius of a Right-Angled Triangle
For a right triangle, the hypotenuse acts as the diameter of the circumcircle. Therefore, the formula becomes:
\(\displaystyle R = \frac{\text{Hypotenuse}}{2}\)
Example: If the sides of a right triangle are 6 cm, 8 cm, and 10 cm, then:
\(\displaystyle R = \frac{10}{2} = 5 \,\text{cm}\)
Circumradius of an Equilateral Triangle
For an equilateral triangle with side length a:
\(\displaystyle R = \frac{a}{\sqrt{3}}\)
Example: For side a = 6:
\(\displaystyle R = \frac{6}{\sqrt{3}} = 2\sqrt{3} \,\text{cm}\)
Steps to Find Circumradius of a Triangle
1. Identify the type of triangle (scalene, right, or equilateral).
2. Use the corresponding formula:
– General: \(\displaystyle \frac{abc}{4\Delta}\)
– Right triangle: \(\displaystyle \frac{\text{Hypotenuse}}{2}\)
– Equilateral triangle: \(\displaystyle \frac{a}{\sqrt{3}}\)
3. Calculate the area if required using Heron’s formula:
\(\displaystyle \Delta = \sqrt{s(s-a)(s-b)(s-c)}\) ,
where \(\displaystyle s = \frac{a+b+c}{2}\) is the semi-perimeter.
4. Substitute values into the formula.
Example Problem
Find the circumradius of a triangle with sides 7 cm, 8 cm, and 9 cm.
Step 1: Find the semi-perimeter:
\(\displaystyle s = \frac{7+8+9}{2} = 12\)
Step 2: Find the area using Heron’s formula:
\(\displaystyle \Delta = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} = 12\sqrt{5}\)
Step 3: Apply the formula:
\(\displaystyle R = \frac{7 \times 8 \times 9}{4 \times 12\sqrt{5}} = \frac{504}{48\sqrt{5}} = \frac{10.5}{\sqrt{5}} \approx 4.69 \,\text{cm}\)
So, the circumradius is approximately 4.69 cm.
Conclusion
Finding the circumradius of a triangle depends on the type of triangle and the available information. The universal formula is \(\displaystyle \frac{abc}{4\Delta}\) , but simpler forms exist for right-angled and equilateral triangles. By mastering this concept, students can solve geometry and trigonometry problems more efficiently.
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Frequently Asked Questions (FAQs)
Ans: The circumradius is the radius of the circle that passes through all three vertices of a triangle.
Ans: No, it depends on the type of triangle – different formulas apply for equilateral, right, and scalene triangles.
Ans: Yes, for right and equilateral triangles, you can directly use simplified formulas.
Ans: It is useful in trigonometry, circle geometry, navigation, and in constructing geometric models.
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