2. Circles

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to a point on the circle is called the radius.

A circle is a two-dimensional shape defined by all points that are the same distance from a center point. This fixed distance is known as the radius. Mathematically, if you know the center of a circle, you can use the radius to locate all the points that make up the circle.

Imagine you have a round pizza. The center of the pizza is the point where you might place a pepperoni. The distance from that pepperoni to the edge of the pizza is the radius. Every point around the edge is same distance from the center, making it a perfect circle.

• Radius (r): Distance from center to any point on the circle.
• Diameter (d): Longest chord, passing through the center (d = 2r).
• Chord: A line segment joining two points on the circle.
• Arc: A part of the circumference.
• Sector: A region bounded by two radii and an arc.
• Segment: A region bounded by a chord and an arc.
• Tangent: A line that touches the circle at exactly one point.

A circle has several important parts:

• Radius: The distance from the center to any point on the circle.
• Diameter: A special chord that passes through the center; it is always twice the length of the radius.
• Chord: Any line connecting two points on the circle.
• Arc: A segment of the circle's circumference.
• Sector: The area enclosed by two radii and the arc between them.
• Segment: The area enclosed by a chord and the arc.
• Tangent: A straight line that touches the circle at just one point, without crossing it.

Picture a ferris wheel! The radius is the distance from the center of the wheel to any seat. The diameter is the longest line that stretches from one side of the wheel to the other, passing through the center. If you were to draw a slice of pie from the center to the edge, that slice is like a sector, while the crust represents an arc.

1. The angle subtended by a diameter is 90°
2. Angles in the same segment are equal
3. The angle at the center is twice the angle at the circumference
4. The perpendicular from the center to a chord bisects the chord
5. Tangents from an external point are equal in length

Theorems about circles help us understand their properties:

1. Any triangle formed by a diameter will always have a right angle (90°) at the circumference.
2. Angles that subtend the same arc are equal.
3. The angle at the circle's center will always be double any angle on the circumference that subtends the same arc.
4. If you draw a line from the center of the circle that is perpendicular to a chord, that line will cut the chord into two equal lengths.
5. Any two lines drawn as tangents from a point outside the circle will be the same length.

Consider the angle you make when you look at a shadow on the ground from a lamp post (the diameter being the light). If you stand exactly below the lamp post (the center) and look at the shadow, your angle to the tip of the shadow forms a right angle with the ground (the diameter). Similarly, if you were at two different points outside a circular pond and cast lines to touch the water, those lines would be equal, just like the tangents.

• Circumference of a circle: 𝐶 = 2𝜋𝑟 • Area of a circle: 𝐴 = 𝜋𝑟² • Length of an arc: Arc length = (θ/360°) × 2𝜋r • Area of a sector: Sector area = (θ/360°) × πr²

There are key formulas in circle geometry that are essential for calculations:

• The circumference (the distance around the circle) can be calculated using the formula: C = 2πr, where r is the radius.
• The area (the space inside the circle) is calculated with: A = πr².
• The arc length for a specific angle θ is found using the formula: Arc length = (θ/360°) × 2πr.
• The area of a sector (a 'slice' of the circle) can be calculated with: Sector area = (θ/360°) × πr².

If you’re baking cookies in the shape of circles, you could calculate how much dough you need for the entire cookie (area) or the edge of the cookie (circumference). If you slice one of those cookies into 'pizza' slices (sectors), you can use the formulas to determine how much dough each slice will require, based on the angle of the slice.

Example 1: A circle has a radius of 7 cm. Find the area and circumference. Solution: $Area = \pi \times 7^2 = 154 \text{ cm}^2 \newline Circumference = 2\pi \times 7 = 44 \text{ cm}$

Example 2: An arc subtends an angle of 60° at the center of a circle with radius 10 cm. Find the arc length. Solution: $Arc length = \frac{60}{360} \times 2\pi\times10 = \frac{1}{6} \times 20\pi \approx 10.47 \text{ cm}$

Example 3: Prove that tangents drawn from an external point to a circle are equal.

Problem-solving with circles often involves applying the formulas and theorems learned:

• In the first example, using the formula for area and circumference allows us to quickly calculate these values for any circle as long as we know the radius.
• The second example shows how to find the length of an arc by applying the angle subtended and the radius, enabling us to calculate lengths of segments of circles.
• The third example deals with proving a property using congruent triangles, demonstrating the connections between circles and geometric principles.

If you're designing a circular garden and want to create a fence around it (circumference), knowing how much area you can plant flowers in (area) is essential! By using these formulas, you can determine the best layout and material needed. In another scenario, if you're setting up a circular sprinkler system, knowing how far the water reaches (arc length) helps you plan the coverage effectively.

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