IB MYP 4-5 Maths- Circle geometry- Study Notes - New Syllabus
IB MYP 4-5 Maths- Circle geometry – Study Notes
Standard
- Circle geometry
IB MYP 4-5 Maths- Circle geometry – Study Notes – All topics
Circle and Its Theorems
A circle is the set of all points in a plane that are at a fixed distance (called the radius) from a fixed point (called the center).
Key Terms:
- Radius: Distance from center to any point on the circle.
- Diameter: A chord passing through the center (twice the radius).
- Chord: A line segment with both endpoints on the circle.
- Arc: Part of the circumference of the circle.
- Sector: Region between two radii and an arc.
- Segment: Region between a chord and an arc.
- Tangent: A straight line touching the circle at exactly one point.
Important Circle Theorems:
| Theorem | Description | |
|---|---|---|
| Angle in a Semicircle |  | An angle subtended by a diameter at the circumference is 90°. |
| Angles in the Same Segment |  | Angles subtended by the same arc at the circumference are equal. |
| Angle at Center vs Circumference |  | The angle subtended at the center of a circle is twice the angle at the circumference subtended by the same arc. |
| Cyclic Quadrilateral |  | Opposite angles in a cyclic quadrilateral add up to 180°. |
| Alternate Segment Theorem |  | The angle between a tangent and a chord equals the angle in the alternate segment. |
| Tangent-Radius Theorem |  | A tangent to a circle is perpendicular to the radius at the point of contact. |
Key Formulas :
1. Circumference of a Circle:
The total distance around the circle.
Formula: \( C = 2\pi r \) or \( C = \pi d \)
Example:
Find the circumference of a circle with radius 7 cm. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
\( C = 2 \times 3.14 \times 7 = 43.96 \,\text{cm} \).
2. Area of a Circle:
The space enclosed within the circle.
Formula: \( A = \pi r^2 \)
Example:
Find the area of a circle with diameter 14 cm. Use \( \pi = 3.14 \).
▶️ Answer/Explanation
Radius \( r = \dfrac{14}{2} = 7 \,\text{cm}\).
Area \( A = 3.14 \times 7^2 = 153.86 \,\text{cm}^2 \).
3. Arc Length:
The length of a part of the circumference corresponding to an angle at the center.
Formula: \( L = \dfrac{\theta}{360^\circ} \times 2\pi r \)
Example:
Find the length of an arc of a circle with radius 14 cm, subtending an angle of 90° at the center.
▶️ Answer/Explanation
\( L = \dfrac{90}{360} \times 2 \times 3.14 \times 14 \).
\( L = 0.25 \times 87.92 = 21.98 \,\text{cm}\).
4. Area of a Sector:
The region bounded by two radii and the arc between them.
Formula: \( \text{Area} = \dfrac{\theta}{360^\circ} \times \pi r^2 \)
Example:
Find the area of a sector of a circle with radius 10 cm and angle 60°.
▶️ Answer/Explanation
\( \text{Area} = \dfrac{60}{360} \times 3.14 \times (10)^2 \).
\( = \dfrac{1}{6} \times 3.14 \times 100 = 52.33 \,\text{cm}^2 \).
Major and Minor Sectors of a Circle
A circle can be divided into two regions by two radii and the arc connecting them. These regions are called sectors.
- Minor Sector: The smaller region enclosed by two radii and the connecting arc.
- Major Sector: The larger region enclosed by the remaining part of the circle and the same two radii.
Relationship:
The sum of the areas of the major and minor sectors equals the total area of the circle.
Formulas:
- Area of Minor Sector: \( \text{Area} = \dfrac{\theta}{360^\circ} \times \pi r^2 \)
- Area of Major Sector: \( \text{Area} = \dfrac{(360^\circ – \theta)}{360^\circ} \times \pi r^2 \)
Example:
A circle has radius 14 cm. Find the area of the minor sector if the central angle is \( 90^\circ \). (Use \( \pi = 3.14 \))
▶️ Answer/Explanation
Step 1: Use formula: \( \text{Area of minor sector} = \dfrac{\theta}{360^\circ} \times \pi r^2 \)
\( = \dfrac{90}{360} \times 3.14 \times (14)^2 \)
\( = \dfrac{1}{4} \times 3.14 \times 196 \)
\( = 153.86 \,\text{cm}^2 \)
Example:
A circle has radius 10 cm. Find the area of the major sector if the central angle is \( 60^\circ \). (Use \( \pi = 3.14 \))
▶️ Answer/Explanation
Step 1: Major angle = \( 360^\circ – 60^\circ = 300^\circ \).
Step 2: Use formula: \( \text{Area of major sector} = \dfrac{300}{360} \times 3.14 \times (10)^2 \)
\( = \dfrac{5}{6} \times 3.14 \times 100 \)
\( = 261.67 \,\text{cm}^2 \)