📐 Circle Theorems Calculator
Calculate angles using circle theorems
Angle in a Semicircle
Any angle inscribed in a semicircle is always a right angle (90°).
Angle = 90°
No inputs needed. Click calculate to see the result.
How to Use This Calculator
Select a Theorem
Choose the circle theorem you want to use from the dropdown menu.
Enter Known Values
If the theorem requires inputs, enter the known angle values. Some theorems (like angle in semicircle) don't need inputs.
Calculate
Click the "Calculate" button to get the result based on the selected theorem.
Circle Theorems and Formulas
1. Angle in a Semicircle
Any angle inscribed in a semicircle is always a right angle.
Angle = 90°
2. Angle at Center vs Angle at Circumference
The angle at the center is twice the angle at the circumference.
Angle at circumference = (Angle at center) / 2
Example: If angle at center = 120°, then angle at circumference = 60°
3. Angles in the Same Segment
Angles in the same segment of a circle are equal.
All angles subtended by same arc are equal
4. Tangent and Radius
The angle between a tangent and a radius at the point of contact is 90°.
Angle = 90°
5. Cyclic Quadrilateral
Opposite angles in a cyclic quadrilateral sum to 180°.
Angle A + Angle C = 180° and Angle B + Angle D = 180°
Sum of all angles = 360°
About Circle Theorems Calculator
The Circle Theorems Calculator helps you calculate angles in circles using fundamental geometric theorems. These theorems are essential for solving circle geometry problems in mathematics.
When to Use This Calculator
- Geometry Problems: Solve circle angle problems in homework and exams
- Education: Learn and verify circle theorem calculations
- Engineering: Apply circle theorems in design and construction
- Architecture: Calculate angles in circular and curved designs
- Verification: Check manual calculations of circle angles
Why Use Our Calculator?
- ✅ Multiple Theorems: Access to 5 essential circle theorems
- ✅ Educational: Shows theorem descriptions and formulas
- ✅ Accurate: Precise mathematical calculations
- ✅ Easy to Use: Simple interface for all skill levels
- ✅ 100% Free: No registration required
Understanding Circle Theorems
Circle theorems are fundamental rules that describe relationships between angles, arcs, and segments in circles. They're derived from Euclidean geometry and are essential for solving circle problems.
- These theorems apply to all circles regardless of size
- They're based on the properties of circles and inscribed angles
- Understanding these theorems helps solve complex geometry problems
- Many theorems are related and can be used together
Real-World Applications
Architecture: Design circular structures, arches, and domes using circle theorem principles.
Engineering: Calculate angles in circular components, gears, and rotating systems.
Navigation: Use circle theorems in GPS calculations and circular path planning.
Frequently Asked Questions
What is an inscribed angle?
An inscribed angle is an angle whose vertex lies on the circle and whose sides are chords of the circle. It's always half the measure of the intercepted arc.
Why is an angle in a semicircle always 90°?
A semicircle subtends 180° at the center. By the angle at center theorem, the inscribed angle is half of this, which is 90°. This is a fundamental property of semicircles.
What is a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided polygon where all vertices lie on a circle. The opposite angles always sum to 180°, making it useful for solving angle problems.
How do I know which theorem to use?
Look at what information you have and what you need to find. If you see a semicircle, use the semicircle theorem. If you have a cyclic quadrilateral, use that theorem. Often, multiple theorems apply to the same problem.
Can I use degrees and radians interchangeably?
This calculator uses degrees. To convert: radians = degrees × (π/180). Most geometry problems use degrees, while calculus problems often use radians.
Are these theorems only for perfect circles?
Yes, these theorems apply specifically to perfect circles. For ellipses or other curves, different geometric principles apply.