Circles: Circumference and Area

⭕ Circles: Circumference and Area

Part 1 of 5 — The Parts of a Circle and the Magic Number π

Topics in This Part

🔑 Key Concept: Every circle hides one special number, π (pi). It connects the distance around a circle to the distance across it — and it's the key to everything in this lesson.

The Parts of a Circle

A circle is the set of all points the same distance from a center point. Three words describe its size:

The circumference is the distance all the way around the circle — like the perimeter of a polygon, but for a curve.

🔑 Key Relationship: The diameter is twice the radius, because it crosses the circle through the center — one radius on each side. $d = 2r \qquad\text{and}\qquad r = \frac{d}{2}$

Radius and Diameter 🧮

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

1) A circle has radius $r = 6$ cm. Its diameter is $d = \,?$ cm. 2) A circle has diameter $d = 20$ in. Its radius is $r = \,?$ in. 3) A pizza is $14$ inches across (that's the diameter). Its radius is $r = \,?$ in.

What Is π (Pi)?

Take any circle, measure all the way around (circumference $C$), then measure straight across (diameter $d$). Divide $C$ by $d$. You always get the same number — about 3.14. That number is π.

$\pi = \frac{C}{d} \approx 3.14$

💡 π never changes. A coin, a clock, a planet's orbit — divide the distance around by the distance across and you always get $\approx 3.14$. That's what makes π so powerful.

Two values you'll use in Grade 7:

Concept Check 🎯

Match Each Term 🔽

Choose the description that fits each part of a circle.

You're Ready to Measure

You now know the building blocks:

• Radius $r$: center to edge.
• Diameter $d = 2r$: across, through the center.
• π $\approx 3.14$: the ratio $\dfrac{C}{d}$ that every circle shares.

🔑 Because $\pi = \dfrac{C}{d}$, multiplying both sides by $d$ gives $C = \pi d$. That one rearrangement is the circumference formula — and it's where Part 2 begins.

⭕ Circles: Circumference and Area

Part 2 of 5 — Finding the Circumference

🔑 The Formula: The circumference is π times the diameter. Because $d = 2r$, you can write it two equal ways: $C = \pi d \qquad\text{or}\qquad C = 2\pi r$

The Circumference Formula

Worked Example: Diameter Given

A circle has diameter $d = 10$ cm. Find the circumference. Use $\pi \approx 3.14$.

$C = \pi d \approx 3.14 \times 10 = 31.4 \text{ cm}$

Worked Example: Radius Given

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

$C = 2\pi r \approx 2 \times 3.14 \times 5 = 31.4 \text{ cm}$

💡 Same answer! A radius of $5$ and a diameter of $10$ describe the same circle, so both formulas give $31.4$ cm. They're really the same formula.

Pick the Right Setup 🔽

For each circle, choose the correct first step. Use $\pi \approx 3.14$.

Compute the Circumference 🧮

Use $\pi \approx 3.14$. Round to the nearest tenth if needed.

1) $d = 6$ m: $\;C = \,?$ m 2) $r = 10$ ft: $\;C = \,?$ ft 3) $d = 100$ cm: $\;C = \,?$ cm

When the Radius Is a Multiple of 7

When the radius or diameter is a multiple of $7$, the fraction $\pi \approx \dfrac{22}{7}$ makes the math come out clean.

Worked Example: $r = 7$ cm

$C = 2\pi r \approx 2 \times \frac{22}{7} \times 7 = 2 \times 22 = 44 \text{ cm}$

The $7$ in the radius cancels the $7$ in the denominator — no decimals needed!

⚠️ Watch your units. Circumference is a length, so its units are plain (cm, m, ft) — not squared. Squared units belong to area, coming in Part 3.

Concept Check 🎯

⭕ Circles: Circumference and Area

Part 3 of 5 — Finding the Area

🔑 The Formula: The area of a circle is π times the radius squared. $A = \pi r^2$ The little $2$ matters — area is always measured in square units.

The Area Formula

$A = \pi r^2 = \pi \times r \times r$

⚠️ Order of operations! Square the radius first, then multiply by π. Do not multiply $\pi \times r$ and then square — that gives the wrong answer.

Worked Example: $r = 5$ cm

$A = \pi r^2 \approx 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ cm}^2$

Step by step:

1. Square the radius: $5^2 = 25$.
2. Multiply by π: $3.14 \times 25 = 78.5$.
3. Units are squared: $\text{cm}^2$.

Concept Check 🎯

Compute the Area 🧮

Use $\pi \approx 3.14$. Square the radius first!

1) $r = 2$ cm: $\;A = \,?$ $(cm^{2})$ 2) $r = 3$ m: $\;A = \,?$ $(m^{2})$ 3) $r = 10$ ft: $\;A = \,?$ $(ft^{2})$

When You're Given the Diameter

The area formula needs the radius. If you're handed the diameter, find the radius first by halving it.

Worked Example: $d = 12$ cm

1. Halve the diameter: $r = \dfrac{12}{2} = 6$ cm.
2. Square it: $6^2 = 36$.
3. Multiply by π: $A = 3.14 \times 36 = 113.04 \text{ cm}^2$.

⚠️ The #1 area mistake: plugging the diameter straight into $A = \pi r^2$. Always cut the diameter in half to get $r$ first. Using $12$ instead of $6$ would give an answer four times too big.

Build the Area Calculation 🔽

A circle has diameter $d = 8$ cm. Step through finding the area. Use $\pi \approx 3.14$.

⭕ Circles: Circumference and Area

Part 4 of 5 — Real-World Problems, Half Circles, and Working Backward

🔑 The skill that matters most: reading a word problem and deciding — do they want the distance around (circumference) or the space inside (area)?

Circumference or Area? Read the Clues

💡 Quick test: If the answer is a length (around the edge), it's circumference. If it's a surface (filling the inside), it's area — and the units are squared.

Around or Inside? 🔽

Decide whether each situation needs circumference or area.

Worked Example: A Circular Pond

A circular pond has a radius of $7$ m.

(a) A path goes all the way around the edge. How long is it? That's the circumference. Using $\pi \approx \dfrac{22}{7}$: $C = 2\pi r \approx 2 \times \frac{22}{7} \times 7 = 44 \text{ m}$

(b) How much water surface does the pond cover? That's the area. Using $\pi \approx \dfrac{22}{7}$: $A = \pi r^2 \approx \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154 \text{ m}^2$

🔑 Same circle, two questions. The path (around) is a length, $44$ m. The surface (inside) is an area, $154$ $m^{2}$. Notice the squared units flag it as area.

Half Circles (Semicircles)

A semicircle is half a circle. Its area is just half the full circle's area:

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

Worked Example: $r = 4$ cm

$A = \frac{1}{2} \times 3.14 \times 4^2 = \frac{1}{2} \times 3.14 \times 16 = \frac{1}{2} \times 50.24 = 25.12 \text{ cm}^2$

💡 For the perimeter of a semicircle, take half the circumference and add the straight diameter across the bottom: $\frac{1}{2}(2\pi r) + d$. The flat edge is part of the boundary too!

Apply It 🧮

Use $\pi \approx 3.14$. Read carefully — around or inside?

1) A round trampoline has radius $5$ ft. How much netting covers the jumping surface? $A = \,?$ $ft^{2}$ 2) A circular running track has diameter $80$ m. How far is one lap around it? $C = \,?$ m 3) A semicircular rug has radius $2$ m. What is its area? $A = \,?$ $m^{2}$

Working Backward 🎯

Sometimes you know the circumference or area and must find the radius or diameter.

⭕ Circles: Circumference and Area

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name the parts of a circle, (2) find circumference with $C = \pi d$ or $C = 2\pi r$, (3) find area with $A = \pi r^2$, and (4) decide which one a word problem needs. Let's put it all together.

Quick Reference

⚠️ Three reminders: (1) Area uses the radius, so halve the diameter first. (2) Square the radius before multiplying by π. (3) Around = length; inside = squared units. Use $\pi \approx 3.14$, or $\dfrac{22}{7}$ when the radius is a multiple of $7$.

Mixed Practice 🧮

Use $\pi \approx 3.14$ (round to the nearest hundredth if needed).

1) $r = 8$ m: circumference $C = \,?$ m 2) $r = 8$ m: area $A = \,?$ $m^{2}$ 3) $d = 4$ cm: area $A = \,?$ $cm^{2}$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.