Circumference and Area of Circles

Circles — Formulas, Applications, and ACT Strategies

In this lesson:

• Compute circumference using both formulas
• Compute area and avoid the diameter trap
• Solve shaded region and composite figure problems

What You Will Learn Today

After this lesson, you will:

1. Compute circumference with or
2. Compute area with
3. Convert between radius and diameter
4. Find missing dimensions from or
5. Solve shaded region and composite problems

Two Formulas for Circle Circumference

These are the same formula because .

Circumference When Given the Radius

A circle has radius 7 cm. Find the circumference.

Step 1: Choose the radius form

Step 2: Substitute and simplify

ACT tip: If answer choices contain , leave your answer as .

Circumference When Given the Diameter

A circle has diameter 20 inches. Find the circumference.

Step 1: Choose the diameter form

Step 2: Substitute and simplify

No conversion needed — the diameter form handles it directly.

Solving for Radius Given Circumference

A circle has circumference feet. Find the radius.

Step 1: Set up the equation

Step 2: Divide both sides by

The cancels — the radius is 15 feet.

Quick Check on Circumference Formulas

A circle has radius 10 cm.

What is its circumference?

Choose the right formula and solve before advancing...

Area Formula Uses Radius Only

The area of a circle measures the space inside:

Critical: This formula requires the radius, not the diameter.

Always write as your first step.

In , only is squared — then multiply by .

Worked Example With Radius Given

A circle has radius 6 cm. Find the area.

Step 1: Substitute into the formula

Step 2: Square the radius, then multiply

Watch Out for the Diameter Trap

A circle has diameter 10 m. Find the area.

First: Convert to radius: m

Then: Apply the formula

Wrong answer: — that's 4 times too large!

Solving for Radius Given the Area

A circle has area square inches. Find the radius.

Step 1: Set up the equation

Step 2: Divide both sides by

Step 3: Take the square root

Quick Check on the Area Formula

A circle has diameter 12 cm. What is its area?

Remember: convert to radius first...

Shaded Regions Use Big Minus Small

Strategy: Shaded area = area of larger shape area of smaller shape

Circle Inscribed in a Square

A circle with radius 5 is inscribed in a square.

Step 1: Square side = diameter =

Step 2: Area of square

Step 3: Area of circle

Shaded area

Annulus: Area Between Two Circles

An annulus has outer radius and inner radius .

Big minus small:

Factored form (faster):

Semicircle Area and Full Perimeter

• Area:
• Perimeter: curved edge + straight edge =

The diameter edge must be included in the perimeter.

Semicircle Perimeter Worked Example Step by Step

A semicircle has radius 4. Find the perimeter.

Curved edge:

Straight edge (diameter):

Total perimeter:

ACT Practice: Find the Semicircle Window

A semicircular window has perimeter inches.

Find the area of the window.

Set up the perimeter equation, solve for r, then find area...

Solution to the Semicircle Window Problem

Step 1: Set up the perimeter equation

Step 2: Factor and solve for

Step 3: Find the area

Key Takeaways and Common Mistakes

• Circumference:
• Area: — always use radius
• Shaded regions: big minus small

Watch out:

• Given diameter? Convert to first
• Don't drop — it's a multiplier
• Semicircle perimeter =

Coming Up Next in Circle Topics

Up next: Arc length and sector area

• Same circle formulas — but for parts of circles
• Central angles determine the fraction you use
• Connects circumference to arc length, area to sectors