Circumradius and Inradius
Radii of triangle circles from trig.
Key idea
Radii of triangle circles from trig.
In depth
R = a/(2 sin A) = abc/(4K) is the circumradius; r = K/s is the inradius, where K is the area and s the semi-perimeter. They link side lengths to the circles touching the triangle.
Worked example
3-4-5 triangle: K = 6, s = 6, r = 1, R = 60/(4·6) = 2.5.
Formula
Real-world uses
- Finding the radius of a circle that fits a triangular garden or a round bridge support
Common mistake
Mixing up the two circles.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Circumradius and Inradius?
Radii of triangle circles from trig. It is a topic at the expert level in Trigonometry.
What is the formula for Circumradius and Inradius?
The key formula is R = a/(2 sinA).
Where is Circumradius and Inradius used in real life?
Real-world examples: Finding the radius of a circle that fits a triangular garden or a round bridge support.
What should I learn before Circumradius and Inradius?
Make sure you are comfortable with Heron from Trigonometry, the previous topic in Trigonometry.
What is a common mistake with Circumradius and Inradius?
Mixing up the two circles.
How can I practise Circumradius and Inradius?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Circumradius and Inradius?
Move on to Trig Substitution in Algebra next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).