A circle is a basic shape in geometry to understand. It is a 2D figure having an area and perimeter. Circle theorems define the relationships among angles, lines, and points in and around a circle.
These theorems are widely used in school mathematics and competitive exams to assess students' geometry skills.
In this blog post, we will be looking at the circle theorems and their definitions, key formulas and easy examples to help you understand and remember them for the long term.
What is a Circle?
A circle is defined as a 2D shape, simple to coin, made up of all points in a plane. The key characteristic of this shape is that its points are at equal distances from the same fixed point called the centre.
In simple words, a circle is a set of all points that are equally distant from its centre.
Basic circle terms
The following are the key terms used to define a circle
- Center: it is a fixed point inside the circle
- Radius (r): the distance from the center to any point on the circle
- Diameter(d): A line passing through the center and joining two points on the circle mathematically, it is twice the radius (d=2r)
- Circumference: It is the boundary or outline of the circle
In a circle, all the radii are of equal length. Wherever you measure the radius, you will find all of them of the same length, irrespective of the point.
A circle has no sides and no corners. It is a closed curve figure to understand. In the real world, you can see the shape in coin form, hanging clock face, and wheel of a car, etc.
Mathematically, in Coordinate Geometry
A circle can be described with respect to a point on the circumference, where (x, y) are the coordinate points, (h,k) are the coordinates of the centre of the circle, and r is the radius.
The circle equation is defined as:
(x-h)2 + (y-k)2 = r2
How is a circle formed?
To form a circle, imagine the line segment bent around until its ends meet. Arrange the loop until it is precisely circular. On paper, a circle is formed by a compass set at a particular radius.
What is the Circle Theorem?
The Circle Theorem is a set of essential rules in geometry that explain the relationships among angles, lines, and points in a circle.
These theorems are deeply focused on how chords, tangents, radii and diameters have relations with each other on a circle. How they have specific angle patterns.
By applying circle theorems, we can calculate unknown angles and lengths without using a protractor or direct measurement.
These theorems have an essential role in solving geometric problems. Once a theorem is known, the solution becomes easier to find. It is straightforward and quick to find.
Before understanding the theorem, a few properties need to be kept in mind about the circle. That circle radius is a fixed measure from any point.
All radii are equal, and the diameter is twice the radius. These properties are the foundation of all circle theorems.
Important Terms related to the Circle Theorem
The following are some basic positions and parts of a circle that every student must know about. The different parts of a circle are explained below.
- Arc: An arc is the connected curve of a circle
- Sector: It is a region bounded by two radii and an arc
- Segment: It is a region bounded by a chord and an arc lying between the chord's endpoints. The segment does not contain the centre.
- Centre: The centre of a circle is the midpoint of a circle
- Chord: A line segment whose endpoints lie on the circle
- Diameter: A line segment having both endpoints on the circle and the largest chord of the circle
- Radius. A line connecting the centre of a circle to any point on the circumference of a circle/circle itself
- Secant: A straight line cutting the circle at two points
- Tangent: It is a straight line touching the circle at a single point
- Subtended Angle: The angle made by two chords meeting at a point on the circumference of a circle is called the subtended angle.
- Inscribed Angle: An angle made from points lying on a circle's circumference. It is the angle subtended at a point on a circle by two given points on the circle. Formed by two chords that have a common endpoint on a circle
- Central Angle: An angle whose vertex is the centre O of a circle and whose sides are radii intersecting the circle in two distinct points.
Common Circle Theorems in Detail
The following are the common circle theorems you can consider in detail.
Alternate Segment Theorem
The theorem is used to find the unknown angle between the tangent and the chord. According to this theorem,
The angle formed between the tangent and the chord at the point of contact is equal to the angle formed by the same chord in the alternate segment.
Let x be the angle formed between the tangent and the chord, according to the figure.
- ∠x = ∠BCE = ∠BAC
Here, ∠BCE is the angle made by the tangent and the chord BC at the point of contact. ∠BAC is the angle made by the same chord BC in the alternate segment.
- ∠y = ∠ABC = ∠ACD
Central Angle Theorem: The angle at the center theorem
According to the central angle theorem,
"The central angle subtended by two points on a circle is always twice the inscribed angle subtended by those points"
Consider a BAC as an inscribed angle, which is an angle made from points lying on the circle circumference. BOC as the central angle, an angle whose vertex is the centre O of a circle and whose sides are radii intersecting the circle in two distinct points B and C
According to the theorem, an inscribed angle ∠BAC = x° is half of the measure of the central angle ∠BOC = 2x°
- ∠BOC = 2 ∠BAC
Angles in the same segment theorem
According to this theorem, angles formed by the same segment region of a circle are equal. Either a major or a minor segment of a circle. For the same segments, angles are also equal ∠BAD=∠BCD.
Angle in a semicircle theorem
An angle in a semicircle measures 90 degrees, and it is always a right angle. In simple words, an angle subtended by a circle's diameter will always be a right angle. Mathematically it is ∠A = ∠B = ∠C = 90°
Chord of a circle theorem
According to the chord theorem of a circle. The perpendicular drawn from the centre of the circle to a chord bisects the chord. That is the angle formed on such a chord at equal and perpendicular angles to each other. Their lengths are also equal.
- ∠OPA = ∠OPB = 90°
- AP=PB
Angles Subtended by Equal Chords Theorem
This concept holds that the two equal chords of a circle subtend equal angles at the centre of the circle. In reverse, if two angles subtended at the centre by two chords are equal, then the chords are of equal length.
According to the diagram, the chords XY and ST are equal. So the angles they form are also the same.
∠XOY = ∠SOT
Cyclic Quadrilateral Theorem
This theory states that the sum of the opposite angles in a cyclic quadrilateral is 180 degrees. A quadrilateral is a four-sided figure whose all vertices lie on a circle. In this figure, ABCD is a cyclic quadrilateral. Mathematically, the theorem is
∠A +∠C =180° and ∠B + ∠D=180°
Tangent of a Circle Theorem
This theory holds that the tangent to a circle is perpendicular to the radius at the point of tangency. In a given diagram, the radius OP is perpendicular to the tangent AB
OP ⊥ AB
∠OPA = ∠OPB = 90°
Conclusion
Circle theorems are required in geometry to comprehend the connections between angles, lines, and points in a circle. Once you understand the fundamentals of concepts and terms of a circle, you will find applying various circle theorems to a trigonometric problem simple.
Since there are the alternate segments, central angle, cyclic quadrilateral, and the tangent, each theory is a logical principle that simplifies the calculation of an angle that would otherwise be challenging to detect.
The theorems minimise the use of actual physical measurements and instead rely on reasoning and mathematical relationships to solve problems faster and more precisely.
Knowing circle theorems not only enhances a student's background in geometry but also improves logical thinking and exam performance.
Circle theorems become an approachable and powerful mathematical topic with consistent practice, clear visuals, and a solid understanding of definitions and formulas.