Circles, Angles, and Angular Movement Chapter Notes | Mathematics for Grade 10 PDF Download

Parts of a Circle

• Circumference: The total length around the edge of a circle. It is calculated using the formulas:
  C = 2πr (where r is the radius)
  C = πd (where d is the diameter).

• Diameter: A straight line passing through the center of the circle, connecting two opposite points on the circumference. It is twice the length of the radius: d = 2r.

• Radius: A line segment from the center of the circle to any point on the circumference.

• Arc: A portion of the circumference of a circle.

• Sector: A pie-shaped region of a circle enclosed by two radii and the arc between them.

• Central Angle: An angle with its vertex at the center of the circle, formed by two radii.

• Standard Position of an Angle: Angles are measured starting from the positive horizontal axis (right side) and moving counterclockwise, unless otherwise specified.

Understanding Pi (π)

Definition of Pi:

• Pi (π) is the ratio of a circle’s circumference to its diameter: π = C/d.
• This ratio is constant for all circles, regardless of size, approximately equal to 3.14 or 22/7 as a fraction.

Key Formulas:

• Circumference: C = πd or C = 2πr.
• Diameter: d = 2r.

Properties:

• Pi is a constant, meaning its value does not change with the size of the circle.
• It is used in calculations involving circumferences, arc lengths, and areas of circles.

Measuring Angles in Degrees

Degrees and Revolutions:

• A full revolution (complete circle) is 360°.
• One degree (1°) is 1/360 of a revolution.

Subdividing Degrees:

• Decimal Notation: Angles can be expressed as decimals, e.g., 130.831°.

Minutes and Seconds:

• 1° = 60 minutes (60').
• 1 minute (1') = 60 seconds (60").
• 1 second (1") = 1/3600 of a degree.

Converting Between Notations

• Multiply the decimal part by 60 to get minutes.
• Multiply the decimal part of minutes by 60 to get seconds.
• Example: 130.831° = 130° + (0.831 × 60)' = 130° + 49.86' = 130° + 49' + (0.86 × 60)" = 130°49'52".

Degree-Minute-Second to Decimal:

• Convert minutes to degrees by dividing by 60.
• Convert seconds to degrees by dividing by 3600.
• Add all parts.
• Example: 28°45'33" = 28 + (45/60) + (33/3600) ≈ 28.759°.

Using Calculators:

• Calculators can convert between decimal degrees and degree-minute-second forms using specific functions.

Radian Measure

Definition of a Radian:

• A radian is the measure of a central angle that subtends an arc equal in length to the circle’s radius.
• Formula: θ = s/r (where θ is the angle in radians, s is the arc length, and r is the radius; s and r must be in the same units).

Calculating Radians:

• Example: For an arc length s = 2 cm and radius r = 5 cm, θ = 2/5 = 0.4 rad.

Concentric Circles:

• Circles with the same center but different radii. The ratio of arc length to radius (θ = s/r) remains constant for a given central angle across concentric circles.

Converting Between Degrees and Radians

Key Relationships:

• A full revolution is 360° or 2π radians: 2π rad = 360°.

• 1 rad = 180°/π ≈ 57.296°.

• 1° = π/180 rad.

Conversion Formulas:

• Radians to Degrees: Multiply radians by 180°/π.
  Example: 2π/3 rad = (2π/3) × (180°/π) = 120°.
• Degrees to Radians: Multiply degrees by π/180°.
  Example: 20° = 20 × (π/180°) = π/9 rad.

Common Conversions:

• 30° = π/6 rad
• 45° = π/4 rad
• 60° = π/3 rad
• 90° = π/2 rad
• 180° = π rad
• 360° = 2π rad

Points to Remember

• The circumference of a circle is C = 2πr or C = πd, where π ≈ 3.14 is a constant ratio of circumference to diameter.
• A diameter is twice the radius (d = 2r), and a radius connects the center to the circumference.
• An arc is a portion of the circumference, and a sector is a pie-shaped area formed by two radii and an arc.
• A central angle has its vertex at the circle’s center, measured counterclockwise from the positive horizontal axis in standard position.
• One revolution = 360° = 2π radians. One degree = 1/360 of a revolution, and one radian ≈ 57.296°.
• Convert degrees to minutes and seconds: 1° = 60', 1' = 60". Decimal degrees can be converted to degree-minute-second form and vice versa.
• A radian is the angle subtending an arc equal to the radius: θ = s/r.
• Convert radians to degrees by multiplying by 180°/π; convert degrees to radians by multiplying by π/180°.
• Concentric circles share the same center, and the radian measure of a central angle is consistent across them.

Difficult Words

• Circumference: The total length around a circle’s edge.
• Diameter: A line through the center connecting two opposite points on the circumference.
• Radius: A line from the center to a point on the circumference.
• Arc: A segment of the circumference.
• Sector: A region bounded by two radii and an arc.
• Central Angle: An angle with its vertex at the circle’s center.
• Pi (π): The constant ratio of a circle’s circumference to its diameter, approximately 3.14.
• Radian: A unit of angle measure where the arc length equals the radius.
• Concentric Circles: Circles with the same center but different radii.

Summary

Circle Properties:

• Circumference: C = 2πr or C = πd.
• Diameter: d = 2r.
• Arc: Part of the circumference.
• Sector: Area enclosed by two radii and an arc.
• Central Angle: Vertex at the circle’s center.

Angle Measurements:

• Degrees: 360° in a revolution, 1° = 1/360 of a revolution.
• Minutes and Seconds: 1° = 60', 1' = 60", 1" = 1/3600°.
• Radians: θ = s/r, where 2π rad = 360°.

Conversions:

• Radians to degrees: Multiply by 180°/π.
• Degrees to radians: Multiply by π/180°.

Applications:

• Calculate circumferences, arc lengths, and angles in practical scenarios like gears, pulleys, and wheels.