Skip to content
mathlab
0
0
0 XP
Lvl 1
KJ
Lesson 8 ~25 min Unit 3 · Measurement & Geometry +85 XP

Area of Circles

Master $A = \pi r^2$ — and always square the RADIUS, never the diameter.

Today's hook: A circular pizza has diameter 30 cm. A square pizza is 20 cm × 20 cm. Which gives you more pizza?
0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom
Think First
warm-up

A circle fits snugly inside a square whose side equals the circle's diameter. Is the circle's area more than half the square's area, less than half, or exactly half? Sketch and estimate.

Record your answer in your workbook.
1
The Big Idea
+5 XP

The area of a circle depends on its radius $r$. The formula is $A = \pi r^2$. You must square the radius, then multiply by $\pi$. If you are given diameter $d$, always halve it first: $r = d \div 2$. Area is in square units (cm², m²). The formula shows that doubling the radius quadruples the area — because $r$ is squared.

r $A = \pi r^2$ Square $r$ first, then $\times \pi$
$$A = \pi r^2$$
Radius, not diameter!
Given $d$: compute $r = d \div 2$ before substituting into $A = \pi r^2$. Forgetting this gives an answer 4× too large.
Order of operations
$A = \pi \times (r^2)$ — square $r$ first, then multiply by $\pi$. Not $(\pi r)^2$.
Square units
Area is in cm², m², mm², etc. — always square units, unlike circumference which uses linear units.
2
What You'll Master
objectives

Know

  • $A = \pi r^2$ (must use radius)
  • From diameter: $r = d/2$ first
  • Reverse: $r = \sqrt{A/\pi}$

Understand

  • Why the formula uses radius not diameter
  • How sector rearrangement derives $A = \pi r^2$
  • Why area scales with $r^2$ (doubling $r$ quadruples $A$)

Can Do

  • Calculate circle area from radius or diameter
  • Find radius when area is given
  • Find area of semicircle and quarter circle
  • Compare areas to solve real-world problems
3
Words You Need
vocabulary
Area of a circleThe amount of flat space enclosed by the circle. Formula: $A = \pi r^2$. In square units.
Radius ($r$)Distance from centre to edge. Must be used in the area formula — halve the diameter if only $d$ is given.
Squaring ($r^2$)Multiplying $r$ by itself. $5^2 = 25$. This is done before multiplying by $\pi$.
SemicircleHalf a circle cut along a diameter. Area $= \frac{1}{2}\pi r^2$.
Quarter circleOne-quarter of a circle (90° sector). Area $= \frac{1}{4}\pi r^2$.
Inverse operationWorking backwards to find $r$ from $A$: $r = \sqrt{A/\pi}$. Use square root to undo squaring.
4
Spot the Trap
heads-up

Classic error: using the diameter in place of the radius — giving an area four times too large.

Wrong ($d = 10$, used as $r$)

$A = \pi \times 10^2 = 100\pi \approx 314$ cm² ✗

Correct ($r = 10 \div 2 = 5$)

$A = \pi \times 5^2 = 25\pi \approx 78.5$ cm² ✓

Rule: Always ask — is this a radius or a diameter? Halve if diameter.

5
Where Does $A = \pi r^2$ Come From?
+5 XP

Slice a circle into many thin sectors (pizza slices) and rearrange them alternating up/down to form a shape like a rectangle. As the number of slices increases, the shape approaches a true rectangle with width $\approx \pi r$ (half the circumference) and height $= r$. So: Area $= \pi r \times r = \pi r^2$.

Slice into sectors → width ≈ πr r Rearranged → near-rectangle
$$A = \pi r \times r = \pi r^2$$
6
Calculating Area from Radius
+5 XP

When radius is given, substitute directly. Critical order: square $r$ first, then multiply by $\pi$. Example: $r = 5$ cm.

Step 1: $r^2 = 5^2 = 25$

Step 2: $A = \pi \times 25 = 25\pi \approx 78.54$ cm²

Step 1: $r^2 = 5^2 = 25$ Step 2: $A = \pi \times 25$ $\approx 78.54$ cm²
$$A = \pi r^2 \quad \text{(square } r \text{ first)}$$
Calculator order
Press: $r$ → $x^2$ → $\times$ → $\pi$ → $=$. Or type $\pi \times r^2$ with brackets.
7
Calculating Area from Diameter
+5 XP

If diameter $d$ is given, halve it first to find $r$, then apply $A = \pi r^2$. Example: $d = 14$ cm.

Step 1: $r = 14 \div 2 = 7$ cm

Step 2: $r^2 = 49$

Step 3: $A = 49\pi \approx 153.94$ cm²

Step 1: $r = 14 \div 2 = 7$ Step 2: $r^2 = 49$ Step 3: $A = 49\pi \approx 153.94$ cm²
$d \xrightarrow{\div 2} r \xrightarrow{r^2} r^2 \xrightarrow{\times\,\pi} A$
8
Finding $r$ from $A$ · Semicircles · Quarter Circles
+5 XP

Finding $r$ from area: $A = \pi r^2 \Rightarrow r^2 = A/\pi \Rightarrow r = \sqrt{A/\pi}$

Semicircle (half circle): $A_{\text{semi}} = \dfrac{1}{2}\pi r^2$

Quarter circle (90° sector): $A_{\text{quarter}} = \dfrac{1}{4}\pi r^2$

$\frac{1}{2}\pi r^2$ $\frac{1}{4}\pi r^2$
$r = \sqrt{A/\pi}$    $A_\text{semi} = \tfrac{1}{2}\pi r^2$    $A_\text{qtr} = \tfrac{1}{4}\pi r^2$

Watch Me Solve It · 3 examples

WE 1 — Area from Radius
+10 XP
Q1
PROBLEM
Find the area of a circle with radius $r = 5$ cm. Round to 2 decimal places.
  1. 1
    Write the formula
    $$A = \pi r^2$$
  2. 2
    Square the radius first
    $r^2 = 5^2 = 25$
  3. 3
    Multiply by $\pi$
    $A = \pi \times 25 = 25\pi \approx 78.54$ cm²
  4. 4
    State with units
    $A \approx 78.54$ cm²
Answer$A = 25\pi \approx 78.54$ cm²
WE 2 — Area from Diameter
+10 XP
Q2
PROBLEM
Find the area of a circle with diameter $d = 14$ cm. Round to 2 decimal places.
  1. 1
    Find the radius
    $r = d \div 2 = 14 \div 2 = 7$ cm
  2. 2
    Square the radius
    $r^2 = 7^2 = 49$
  3. 3
    Multiply by $\pi$
    $A = \pi \times 49 = 49\pi \approx 153.94$ cm²
Answer$A = 49\pi \approx 153.94$ cm²
WE 3 — Find Radius from Area
+10 XP
Q3
PROBLEM
A circle has area $A = 78.5$ cm². Find the radius to 1 decimal place.
  1. 1
    Rearrange the formula
    $r^2 = A \div \pi = 78.5 \div \pi \approx 24.99$
  2. 2
    Take the square root
    $r = \sqrt{24.99} \approx 5.0$ cm
  3. 3
    Check
    $A = \pi \times 5^2 = 25\pi \approx 78.54$ cm² ✓
Answer$r \approx 5.0$ cm
9
Common Pitfalls
heads-up
Using Diameter Instead of Radius
Mistake: $A = \pi \times 10^2 = 314$ cm² when $d = 10$. The radius is only 5.
Fix: $r = 5$, $A = \pi \times 25 \approx 78.5$ cm². Off by ×4 if you forget to halve.
Squaring $\pi$ Instead of $r$
Mistake: $A = (\pi r)^2$ or $A = \pi^2 r$.
Fix: Only $r$ is squared: $A = \pi \times r^2$. $\pi$ is not squared.
Forgetting the Fraction for Semicircles
Mistake: $A_\text{semi} = \pi r^2$ (full circle area instead of half).
Fix: $A_\text{semi} = \frac{1}{2}\pi r^2$. A semicircle is exactly half the full circle area.
Copy Into Your Books
Full circle: $A = \pi r^2$ — use RADIUS, not diameter
From diameter: $r = d \div 2$ first, then $A = \pi r^2$
Semicircle: $A = \frac{1}{2}\pi r^2$   Quarter circle: $A = \frac{1}{4}\pi r^2$
Find radius: $r = \sqrt{A \div \pi}$

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Area of Circles
4 problems

Set a timer for 4 minutes. Show all working.

  1. 1 Find $A$ for $r = 3$ cm.

    $A = \pi \times 9 = 9\pi \approx 28.27$ cm²

  2. 2 Find $A$ for $d = 10$ cm. (Halve first!)

    $r = 5$. $A = \pi \times 25 = 25\pi \approx 78.54$ cm²

  3. 3 Find $r$ when $A = 200$ cm².

    $r = \sqrt{200/\pi} \approx \sqrt{63.66} \approx 7.98 \approx 8$ cm

  4. 4 Hook: circular pizza $d = 30$ cm vs square $20 \times 20$ cm. Which is bigger and by how much?

    Circle: $r = 15$. $A = \pi \times 225 \approx 706.86$ cm². Square: $A = 400$ cm². Circle wins by $\approx 307$ cm²!
Complete in your workbook.

Quick Check · 5 questions

1
What is the formula for the area of a circle?
+10 XP
2
Circle with $r = 3$ cm. Find its area.
+10 XP
3
Circle with $d = 10$ cm. Find its area.
+10 XP
4
Circular pizza $d = 30$ cm vs square pizza $20 \times 20$ cm. Which has more area?
+10 XP
5
$A = 200$ cm². Find $r$ to the nearest cm.
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
ApplyMedium3 MARKS

Q6. A circular pool has diameter 8 m. Find its area. Calculate the cost to tile the bottom at $45 per m².

Show full working in your book.
UnderstandEasy2 MARKS

Q7. A semicircle has radius 6 cm. Find its area. Round to 2 decimal places.

Show full working in your book.
ReasonHard4 MARKS

Q8. A circle has area 113.1 cm². (a) Find the radius. (b) Find the circumference. (c) A fence around the circle costs $12 per metre. Find the total cost.

Show full working in your book.
Comprehensive Answers

MC: 1-A, 2-C, 3-B, 4-A, 5-B

Q6: $r = 4$ m. $A = \pi \times 16 = 16\pi \approx 50.27$ m². Cost $= 50.27 \times 45 \approx \$2\,262$.

Q7: $A_\text{semi} = \frac{1}{2} \times \pi \times 36 = 18\pi \approx 56.55$ cm².

Q8: (a) $r = \sqrt{113.1/\pi} \approx \sqrt{36} = 6$ cm. (b) $C = 2\pi \times 6 = 12\pi \approx 37.70$ cm. (c) Cost $= 37.70 \times 12 \approx \$452.40$.

Stretch Challenge · +25 XP

Circular Garden Path

A circular garden has radius 5 m. It is surrounded by a path 1 m wide. Find the area of the path only. (Hint: outer circle minus inner circle.)

Reveal solution

Outer radius $= 6$ m. $A_\text{outer} = 36\pi$ m². Inner radius $= 5$ m. $A_\text{inner} = 25\pi$ m². Path area $= 36\pi - 25\pi = 11\pi \approx 34.56$ m².

R
Quick Review

Full circle

$A = \pi r^2$ (radius squared)

From diameter

$r = d/2$ first, then $A = \pi r^2$

Find radius

$r = \sqrt{A/\pi}$

Semicircle

$A = \frac{1}{2}\pi r^2$

Quarter circle

$A = \frac{1}{4}\pi r^2$

Annulus (ring)

$A = \pi R^2 - \pi r^2$

Your Badges

0 of 6
First Steps
3-Day Streak
3 in a Row
Lesson Ace
Stretch Seeker
Daily Warrior
DAILY CHALLENGE
Area Speed Round
Circle with $d = 12$ cm. Find its area in under 60 seconds for double coins!
NEXT UP
Lesson 9 — Surface Area of Prisms and Cylinders
0/5
Finish this lesson's Practice MCs to unlock!

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page · Unit quiz