Circumference and Area of Circles

Circles — Formulas and Applications

In this lesson:

• Compute circumference from a radius or diameter
• Compute area and avoid the diameter trap
• Solve shaded regions, semicircles, and quarter circles

What You Will Learn Today

1. Compute and from a radius or diameter, converting to
2. Answer in terms of or as decimals
3. Find a missing radius or diameter from or
4. Area and perimeter of semicircles and quarter circles; area of shaded regions, annuli

Circumference Starts With Pi as a Ratio

First, two quick checks: a circle with has ; and .

Walk around a circular track: that distance is the circumference.

Solve for :

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

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

Answer Format: Keep Pi or Convert

• Choices contain : keep it. .
• Choices are decimals: multiply by 3.14159. .
• is a factor, not a label: , but .

Area Measures the Space Inside

• Circumference is the distance around; area is the space inside
• Count unit squares: area grows with
• Doubling makes the area 4 times as large

Area Formula Uses Radius Only

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

Spot the Error: The Diameter Trap

A circle has diameter 10 m. A student writes .

The error: 10 is the diameter, not the radius.

Fix: , so .

is 4 times too large.

Solving for the Radius or Diameter From Area

Area in terms of : , so and in.

If the diameter is asked: in.

Area as a plain number:

Circumference Versus Area at

• : a length, with to the first power
• : square units, with

The exponent on tells you the dimension.

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: and

Shaded area

Square Inscribed in a Circle

• Recall: diagonal (Pythagorean theorem)
• , so
• Shaded

Annulus: Area Between Two Circles

An annulus has outer radius and inner radius .

Big minus small:

Factored form (faster):

Semicircle Area and Full Perimeter

Radius 4:

• Area:
• Perimeter:

Include the straight diameter edge.

Quarter Circle Area and Perimeter

Radius 6. How many straight edges?

• Area:
• Perimeter:

Multi-Step: The Semicircular Window Problem

A semicircular window has perimeter in. Find its area.

Key Takeaways and Common Mistakes

• Circumference:
• Area: , always with the radius
• Shaded regions: big minus small

Watch out:

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

Coming Up Next in Circle Topics

Up next: Central and inscribed angles

• Angles whose vertex is the center of the circle
• Angles whose vertex is on the circle
• How each angle relates to the arc it cuts off