Circle Basics

⭕ Circle Basics

Part 1 of 5 — The Parts of a Circle

Topics in This Part

🔑 Key Concept: A circle is the set of all points that are the same distance from a single fixed point. That fixed point is the center, and that fixed distance is the radius. Every formula in this lesson grows from those two ideas.

Center, Radius, and Diameter

A circle is named by its center. A circle with center $O$ is called "circle $O$" (written $\odot O$).

The diameter is a chord that passes through the center, so it is always twice the radius:

$d = 2r \qquad\Longleftrightarrow\qquad r = \dfrac{d}{2}$

Examples

💡 The word radius comes from the Latin for "spoke of a wheel." Like spokes, every radius of the same circle has the same length.

Radius and Diameter 🧮

Use $d = 2r$ and $r = \dfrac{d}{2}$.

1) A circle has radius $r = 8$. Its diameter is $d = \,?$ 2) A circle has diameter $d = 24$. Its radius is $r = \,?$ 3) A circle has radius $r = 4.5$. Its diameter is $d = \,?$ (decimal is fine)

Chords, Secants, and Tangents

Different lines and segments interact with a circle in different ways:

The single point where a tangent touches is the point of tangency.

🔑 Diameter = longest chord. Every diameter is a chord, but most chords are not diameters. A chord only becomes a diameter when it runs through the center.

⚠️ Don't confuse a chord (a segment with both ends on the circle) with a secant (a full line that passes through the circle). A secant contains a chord.

Concept Check 🎯

Arcs and Semicircles

An arc is a portion of the circle itself — a curved piece of the boundary, not a straight segment.

• A semicircle is exactly half the circle, cut off by a diameter (an arc of $180°$).
• A minor arc is smaller than a semicircle (less than $180°$).
• A major arc is larger than a semicircle (more than $180°$).

The whole way around a circle measures $360°$, so:

$\text{semicircle} = 180°, \qquad \text{full circle} = 360°$

💡 We measure arcs in degrees (how much of the $360°$ they cover) and we'll measure their length in Part 4. Two different ideas — same arc.

You now know the vocabulary of a circle. In Part 2 we measure the distance around it.

Match the Vocabulary 🔽

Choose the term that fits each description.

⭕ Circle Basics

Part 2 of 5 — Circumference

🔑 The Idea: The circumference is the distance once around the outside of a circle — its "perimeter." Amazingly, that distance is always the diameter multiplied by the same magic number, $\pi$.

Meet $\pi$ (Pi)

Divide any circle's circumference by its diameter and you always get the same number:

$\pi = \dfrac{C}{d} \approx 3.14159\ldots \approx 3.14$

Because $\pi = \dfrac{C}{d}$, multiplying both sides by $d$ gives the circumference formulas:

$C = \pi d \qquad\text{or, since } d = 2r,\qquad C = 2\pi r$

💡 $\pi$ is irrational — its decimals never end or repeat. For exact answers we leave $\pi$ in the answer (like $10\pi$); for decimals we approximate with $\pi \approx 3.14$.

Worked Examples

Example 1 — given the radius

A circle has radius $r = 5$. Find the circumference.

$C = 2\pi r = 2\pi(5) = 10\pi \approx 10(3.14) = 31.4$

Example 2 — given the diameter

A circle has diameter $d = 10$. Find the circumference.

$C = \pi d = \pi(10) = 10\pi \approx 31.4$

✅ Notice: $r = 5$ and $d = 10$ describe the same circle, so both give $C = 10\pi$. Whether you start from $r$ or $d$, you reach the same answer.

Example 3 — exact answer

A circle has radius $r = 3$. The exact circumference is:

$C = 2\pi(3) = 6\pi$

Concept Check 🎯

Find the Circumference 🧮

Use $\pi \approx 3.14$ and round to two decimal places where needed.

1) $r = 5$. Circumference $C = \,?$ (use $\pi \approx 3.14$) 2) $d = 6$. Circumference $C = \,?$ (use $\pi \approx 3.14$) 3) $r = 10$, exact answer. $C = \,?\,\pi$ (enter just the number before $\pi$)

Working Backwards

If you know the circumference and want the diameter or radius, just undo the multiplication.

$d = \dfrac{C}{\pi}, \qquad r = \dfrac{C}{2\pi}$

Example

A circular track has circumference $C = 62.8$ m. Find its diameter (use $\pi \approx 3.14$).

$d = \dfrac{C}{\pi} = \dfrac{62.8}{3.14} = 20 \text{ m}$

⚠️ Watch the formula match. Dividing $C$ by $\pi$ gives the diameter; dividing by $2\pi$ gives the radius. Mixing these up is the most common circumference error.

In Part 3 we move from the distance around a circle to the space inside it.

Choose the Right Move 🔽

A circle has circumference $C = 31.4$ (use $\pi \approx 3.14$).

⭕ Circle Basics

Part 3 of 5 — Area

🔑 The Idea: Area is the amount of flat space inside a circle. It depends on the radius squared — so doubling the radius makes the area four times as big.

The Area Formula

$A = \pi r^2$

The radius is squared, and the units are square units ($cm^{2}$, $in^{2}$, $m^{2}$).

⚠️ Radius, not diameter! If you're given the diameter, first cut it in half to get $r$, then square it. Forgetting to halve first is the #1 area mistake.

💡 Circumference uses $r$ to the first power ($2\pi r$); area uses $r$ to the second power ($\pi r^2$). The little exponent tells you which one you're finding.

Worked Examples

Example 1 — given the radius

$r = 5$:

$A = \pi r^2 = \pi (5)^2 = 25\pi \approx 25(3.14) = 78.5 \text{ square units}$

Example 2 — given the diameter

$d = 12$. First halve it: $r = \dfrac{12}{2} = 6$. Then:

$A = \pi (6)^2 = 36\pi \approx 113.1 \text{ square units}$

✅ Check the trap: if you forgot to halve and used $r = 12$, you'd get $144\pi$ — four times too big. Always halve the diameter first.

Example 3 — exact answer

$r = 4$:

$A = \pi(4)^2 = 16\pi$

Concept Check 🎯

Find the Area 🧮

Give exact answers as a number times $\pi$ (enter only the number before $\pi$).

1) $r = 3$. Area $= \,?\,\pi$ 2) $r = 7$. Area $= \,?\,\pi$ 3) $d = 20$ (so $r = 10$). Area $= \,?\,\pi$

Half Circles (Semicircles)

A semicircle is half a circle, so its area is half the full-circle area:

$A_{\text{semicircle}} = \dfrac{1}{2}\,\pi r^2$

Example

A semicircle has radius $r = 6$:

$A = \dfrac{1}{2}\,\pi (6)^2 = \dfrac{1}{2}(36\pi) = 18\pi \approx 56.5 \text{ square units}$

💡 The same "take a fraction of the whole" idea — half here — powers sectors in Part 4, where the fraction can be any slice of the $360°$ pie.

Next up: slicing the circle into wedges and arcs.

Semicircle Reasoning 🔽

A semicircle has radius $r = 4$.

⭕ Circle Basics

Part 4 of 5 — Arcs, Sectors & Central Angles

🔑 Big Idea: A central angle opens from the center and cuts out a slice. The fraction of $360°$ that the angle covers is the same fraction of the circumference (arc length) and of the area (sector area).

Central Angles and the "Fraction" Trick

A central angle has its vertex at the center. Its measure equals the measure of the arc it cuts off.

The key to everything in this part is one fraction:

$\text{fraction of the circle} = \dfrac{\text{central angle}}{360°}$

💡 Once you know the fraction, multiply it by the full circumference for arc length, or by the full area for sector area. Same fraction, two uses.

Arc Length

An arc length is the distance along the curved edge — a fraction of the whole circumference:

$\text{arc length} = \dfrac{\theta}{360°} \cdot 2\pi r$

where $\theta$ is the central angle.

Example

A circle has radius $r = 8$ and a central angle of $\theta = 90°$. Find the arc length.

$\text{arc} = \dfrac{90}{360}\cdot 2\pi(8) = \dfrac{1}{4}\cdot 16\pi = 4\pi \approx 12.57$

✅ Sanity check: a $90°$ arc is one quarter of the way around, so it should be one quarter of the circumference $16\pi$. Indeed $\frac{1}{4}(16\pi) = 4\pi$. ✓

Concept Check 🎯

Sector Area

A sector is a "pizza slice" — the region between two radii and an arc. Its area is the same fraction of the total area:

$\text{sector area} = \dfrac{\theta}{360°} \cdot \pi r^2$

Example

A circle has radius $r = 6$ and a $120°$ sector. Find the sector area.

$\text{sector} = \dfrac{120}{360}\cdot \pi(6)^2 = \dfrac{1}{3}\cdot 36\pi = 12\pi \approx 37.70$

⚠️ Pick the right "whole." Arc length uses the circumference $2\pi r$; sector area uses the area $\pi r^2$. The fraction $\frac{\theta}{360}$ is identical — only the thing you multiply by changes.

Arcs and Sectors 🧮

Give exact answers as a number times $\pi$ (enter only the number before $\pi$).

1) $r = 8$, $\theta = 90°$. Sector area $= \,?\,\pi$ 2) $r = 4$, $\theta = 45°$. Sector area $= \,?\,\pi$ 3) $r = 10$, $\theta = 180°$. Arc length $= \,?\,\pi$

A Reliable 3-Step Routine

Every arc-length or sector-area problem reduces to the same three moves:

1. Fraction: compute $\dfrac{\theta}{360°}$ and simplify it.
2. Whole: find the matching "whole" — circumference $2\pi r$ for an arc, area $\pi r^2$ for a sector.
3. Multiply: fraction × whole.

💡 If you keep the fraction simplified (like $\frac{1}{6}$ instead of $\frac{60}{360}$), the final multiplication is almost always clean. Let's run all three steps on one circle.

Build the Sector 🔽

A circle has radius $r = 6$ and a central angle of $\theta = 60°$.

⭕ Circle Basics

Part 5 of 5 — Equation of a Circle & Mastery Check

🔑 Last Idea: Put a circle on the coordinate plane and its definition ("all points a fixed distance $r$ from the center") becomes a clean algebraic equation built from the distance formula.

The Standard Equation

A circle with center $(h, k)$ and radius $r$ has the equation:

$(x - h)^2 + (y - k)^2 = r^2$

Read it carefully — the signs flip, and the right side is $r^2$ (not $r$):

⚠️ Sign flip: $(x - h)$ means the center's $x$ is $+h$. So $(x + 5)^2 = (x - (-5))^2$ gives $h = -5$. And take the square root of the right side to get the radius — $r^2 = 25$ means $r = 5$, not $25$.

Worked Examples

Example 1 — write the equation

Center $(2, -3)$, radius $5$. Substitute $h = 2$, $k = -3$, $r = 5$:

$(x - 2)^2 + (y - (-3))^2 = 5^2 \;\Longrightarrow\; (x-2)^2 + (y+3)^2 = 25$

Example 2 — read off center and radius

Given $(x-1)^2 + (y-4)^2 = 36$:

$\text{center } (1, 4), \qquad r = \sqrt{36} = 6$

✅ Check: plug the center's own coordinates? No — plug a point you expect on the circle. The point $(7, 4)$ is $6$ units right of the center: $(7-1)^2 + (4-4)^2 = 36 + 0 = 36$ ✓.

Read the Circle 🔽

Use $(x-h)^2 + (y-k)^2 = r^2$.

Quick Reference — Everything in One Table

⚠️ Two traps to remember: halve the diameter before squaring for area, and take the square root of $r^2$ to read a radius off an equation. Keep those straight and the rest is bookkeeping.

Center and Radius 🧮

Given $(x+5)^2 + (y-2)^2 = 9$:

1) The center's $x$-coordinate $h = \,?$ 2) The center's $y$-coordinate $k = \,?$ 3) The radius $r = \,?$

Putting It All Together

You can now move fluently between a circle's radius, diameter, circumference, area, its arcs and sectors, and its equation on the plane. The final two checks mix every part of the lesson at once.

💡 When a problem gives you the diameter, jot down the radius immediately — most formulas (area, circumference via $2\pi r$, arc length, sector area) are built on $r$, so converting first prevents the most common slips.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.