Circle Area & Circumference
Circle Formulas
Draw a line to match each formula to its name.
Calculate Circumference
Use π ≈ 3.14. Round to 1 decimal place.
Radius = 5 cm
Diameter = 12 m
Radius = 7 cm
Calculate Area
Use A = πr². Use π ≈ 3.14.
Radius = 4 cm
Radius = 6 m
Diameter = 10 cm
Radius = 3 mm
Find the Radius from Area
Use r = √(A ÷ π).
A = 78.5 cm²
A = 113 m²
Circle vs Square — Larger Area?
Compare areas. Sort each.
Circle Problems
Show all working. Use π ≈ 3.14.
A circular pizza has diameter 30 cm. Find its area and circumference.
A sprinkler waters a circular area with radius 8 m. What area of lawn is watered?
A wheel has circumference 188.4 cm. What is its diameter?
Arc Length
An arc is a fraction of the circumference. Arc length = (θ/360) × 2πr. Use π ≈ 3.14.
r = 6 cm, angle = 90°
r = 10 m, angle = 180°
r = 5 cm, angle = 60°
Area of a Sector
A sector is a 'pizza slice'. Area = (θ/360) × πr². Use π ≈ 3.14.
r = 6 cm, angle = 90°
r = 10 m, angle = 180°
r = 4 cm, angle = 45°
Annulus (Ring) Area
Annulus area = π(R² − r²) where R = outer radius, r = inner radius. Use π ≈ 3.14.
R = 7 cm, r = 4 cm
R = 10 m, r = 8 m
Real-World Circle Problems
Draw a diagram for each problem. Show all working. Use π ≈ 3.14.
A bicycle wheel has diameter 70 cm. How many complete rotations does it make when travelling 1 km? (1 km = 100 000 cm)
A circular garden bed has radius 5 m. A path of width 1 m surrounds it. Calculate the area of the path only.
Match Circle Calculation to Answer
Use π ≈ 3.14. Draw a line to match each circle problem to its answer.
Circumference Practice — Set A
Calculate the circumference. Use π ≈ 3.14. Round to 1 d.p.
r = 3 cm
r = 10 m
d = 14 mm
d = 9 m
Area Practice — Set A
Calculate the area. Use π ≈ 3.14. Round to 2 d.p.
r = 4 cm
r = 9 m
d = 20 cm
d = 7 m
Circle Problems — Set B
Show all working. Use π ≈ 3.14.
A circular pool has radius 6 m. What is its circumference? What is its area?
A roundabout has diameter 24 m. What is its circumference? How much fencing is needed for the edge?
A circular mirror has area 200.96 cm². What is its radius? What is its diameter?
Arc Length Practice
Use arc length = (θ/360) × 2πr. Use π ≈ 3.14.
r = 10 cm, θ = 90°
r = 6 m, θ = 120°
r = 5 cm, θ = 180°
r = 8 m, θ = 270°
Sector Area Problems
Use sector area = (θ/360) × πr². Show all working. Use π ≈ 3.14.
A clock face has radius 15 cm. What is the area of the sector between 12 o'clock and 3 o'clock (90°)?
A pie chart shows 60° for one category. The pie has radius 8 cm. What is the area of that sector?
A sprinkler rotates 240° and reaches 5 m. What area does it water?
Classify Circle Calculations
Sort each calculation into the correct category.
Annulus (Ring) Problems
Annulus area = π(R² − r²). Show all working. Use π ≈ 3.14.
A circular path surrounds a garden. The garden has radius 4 m. The path is 2 m wide. Find the area of the path alone.
A washer has outer diameter 20 mm and inner diameter 10 mm. Find its area.
A running track surrounds a rectangular field. The track's inner radius is 36 m and outer radius is 43 m. Find the area of just the curved end sections (two semicircles).
Circumference Practice — Set B
Use exact π (leave π in answer) OR calculate with π ≈ 3.14159. Give both forms.
r = 8 cm. Give circumference as a multiple of π, then as a decimal.
d = 15 m. Give circumference as a multiple of π, then as a decimal.
r = 1.5 m. Give circumference as a multiple of π, then as a decimal.
Area Practice — Set B
Give the exact area as a multiple of π, then calculate using π ≈ 3.14.
r = 5 cm. Area in exact form and decimal.
d = 16 m. Area in exact form and decimal.
r = 2.5 cm. Area in exact form and decimal.
Multi-Step Circle Problems
Draw a diagram and show all working. Use π ≈ 3.14.
A wheel has radius 35 cm. How many complete rotations does it make travelling 220 m? (Hint: 1 rotation = 1 circumference. Convert metres to centimetres.)
A circular fountain has radius 3 m. A square tile (30 cm × 30 cm) costs $4. Estimate how many tiles are needed to pave the base of the fountain and find the cost.
Composite Shapes with Circles
Calculate the area and/or perimeter of each composite shape. Show all working.
A rectangle 10 cm × 6 cm has a semicircle of diameter 6 cm attached to one end. Find the total area.
A square of side 10 cm has a circular hole of diameter 6 cm cut from its centre. Find the remaining area.
A running track consists of a rectangle 80 m × 40 m with two semicircles (diameter 40 m) at each end. Find the total perimeter (lane 1 distance).
Circle on a Coordinate Grid
A circle has centre (0, 0) and passes through the point (3, 4).
Use the distance formula to find the radius of the circle.
Find the exact circumference and area of this circle.
Does the point (0, −5) lie on, inside, or outside the circle? Explain.
Real-World Application — Sports
Use circle formulas to answer these sports problems. Show all working. Use π ≈ 3.14.
An athletics shot put circle has diameter 2.135 m. Find the circumference and area.
A basketball hoop has inner diameter 45.7 cm. A ball has circumference 75 cm. Will the ball fit through the hoop? Show reasoning.
Maximum Area Investigation
A farmer has 100 m of fencing to make a pen. Investigate different shapes.
If the pen is a square, what are its dimensions and area?
If the pen is a circle, what is its radius and area?
Which shape gives more area? By how much? What does this tell you about circles?
Cylinder Base — Volume Connection
A cylinder has the same base as a circle with radius r and height h.
A cylinder has base radius 5 cm and height 12 cm. Find the base area, then the volume.
A can of soup has radius 4 cm and holds 603 mL. How tall is the can? (1 mL = 1 cm³)
Pi Investigation
Use a calculator with the π button for this investigation.
Calculate the circumference of the Earth (radius ≈ 6 371 km) to the nearest km.
If you walked at 5 km/h non-stop, how many years would it take to walk around the Earth?
The Moon orbits Earth at a radius of approximately 384 400 km. What distance does it travel in one orbit?
Arc and Sector — Problem Solving
Show all working. Use π ≈ 3.14.
A sector has arc length 20 cm and radius 8 cm. Find the angle of the sector.
A sector has area 75.4 cm² and radius 10 cm. Find the angle of the sector.
Design Problem — Logo
A company logo consists of a large circle (r = 10 cm) with four smaller circles (r = 5 cm) arranged symmetrically inside.
Find the total area of the four small circles.
Find the area inside the large circle but outside the small circles.
What percentage of the large circle is covered by the small circles?
Error Analysis
Find and correct the errors in this student's working.
Student: 'Circle with d = 10. Area = π × 10² = 100π ≈ 314 cm².' What is wrong? Give the correct answer.
Student: 'Circle with r = 6. Circumference = π × 6² = 36π ≈ 113 cm.' What is wrong? Give the correct answer.
Reflection and Summary
Answer these questions to summarise what you've learned about circles.
Write the two formulas for circumference and the formula for area. Give an example calculation for each.
Name two real-world situations where you need to calculate circumference and two where you need area.
What is the difference between arc length and circumference? Between sector area and circle area?
Circles in the Real World
Measure and calculate using circular objects at home.
- 1Measure the diameter of 5 circular objects (plate, cup, clock, coin, tin). Calculate the circumference and area of each. Record your results in a table.
- 2Research: how far does a car tyre travel per revolution? (Measure or look up a standard tyre diameter.) How many revolutions does it make on a 10 km journey?
- 3Cut a circle out of paper. Fold it into eighths (fold in half three times). Each fold creates a sector. Estimate the sector angle and area of one piece.