How to Find the Area and Circumference of Circles

How to Find the Area and Circumference of Circles

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A circle is described by its radius and diameter. The radius goes from the center to the circle, and the diameter is twice the radius.

Core Formulas

  • Diameter: \(d=2r\).
  • Circumference: \(C=2\pi r=\pi d\).
  • Area: \(A=\pi r^2\).
  • Arc length for a central angle \(\theta\): \(\frac{\theta}{360^\circ}\cdot 2\pi r\).
  • Sector area: \(\frac{\theta}{360^\circ}\cdot \pi r^2\).

Worked Example

If \(r=6\), then \(C=2\pi(6)=12\pi\) and \(A=\pi(6^2)=36\pi\).

Reference Diagram

Circumference of Circles

Video Lesson

Original Practice Figures

These saved figures are kept with the lesson for continuity. The exercise text below gives all needed measurements.

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Area and Circumference of Circles

Think of this lesson as more than a rule to memorize. Area and Circumference of Circles is about shape relationships, formulas, and units. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Sketch or label the shape.
  • Decide whether the question asks for length, area, volume, or surface area.
  • Substitute values into the matching formula.
  • Keep units squared for area and cubed for volume.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) Find the circumference of a circle with radius 4 cm.

2) Find the area of a circle with radius 5 in.

3) Find the circumference of a circle with diameter 18 ft.

4) Find the area of a circle with diameter 14 m.

5) A circle has circumference \(20\pi\) yd. Find its radius.

6) A circle has area \(81\pi\) square cm. Find its radius.

7) Find both area and circumference for a circle with radius 9.

8) A bike wheel has diameter 26 in. How far does it travel in one rotation?

9) A circular table has radius 3 ft. Find its area.

10) A circle has diameter 40 mm. Find its area.

11) Find the arc length of a \(90^\circ\) arc in a circle of radius 8.

12) Find the sector area of a \(60^\circ\) sector in a circle of radius 12.

13) A semicircle has radius 10. Find its arc length.

14) A quarter circle has radius 6. Find its sector area.

15) A circle has area \(144\pi\). Find its circumference.

16) A circle has circumference \(34\pi\). Find its area.

17) A circular track has radius 50 m. How far is 3 laps?

18) A pizza has diameter 16 in. What is the area of one of 8 equal slices?

19) A \(120^\circ\) sector has radius 9 cm. Find both arc length and sector area.

20) The circumference of a circle is \(18\pi\). Find the area of a circle with twice that radius.

 
1) Use \(C=2\pi r\).
\(C=2\pi(4)=8\pi\) cm.
2) Use \(A=\pi r^2\).
\(A=\pi(5)^2=25\pi\) square in.
3) Use \(C=\pi d\).
\(C=\pi(18)=18\pi\) ft.
4) First find radius: \(r=14/2=7\).
Area \(A=\pi(7)^2=49\pi\) square m.
5) Circumference \(C=2\pi r\).
\(20\pi=2\pi r\), so \(r=10\) yd.
6) Area \(A=\pi r^2\).
\(81\pi=\pi r^2\), so \(r^2=81\).
\(r=9\) cm.
7) Circumference: \(C=2\pi(9)=18\pi\).
Area: \(A=\pi(9)^2=81\pi\).
8) One rotation covers one circumference.
\(C=\pi d=26\pi\) in.
9) Use \(A=\pi r^2\).
\(A=\pi(3)^2=9\pi\) square ft.
10) The radius is \(40/2=20\) mm.
Area \(=\pi(20)^2=400\pi\) square mm.
11) Arc length is \(\frac{90}{360}\cdot2\pi(8)\).
This is \(\frac14\cdot16\pi=4\pi\) units.
12) Sector area is \(\frac{60}{360}\cdot\pi(12)^2\).
This is \(\frac16\cdot144\pi=24\pi\) square units.
13) A semicircle is \(180^\circ\), half the circumference.
Arc length \(=\frac12(2\pi(10))=10\pi\) units.
14) A quarter circle is \(90^\circ\).
Sector area \(=\frac14\pi(6)^2=9\pi\) square units.
15) From \(A=\pi r^2\), \(r^2=144\), so \(r=12\).
Then \(C=2\pi(12)=24\pi\).
16) Since \(C=\pi d\), the diameter is \(34\), so \(r=17\).
Area \(=\pi(17)^2=289\pi\).
17) One lap is \(C=2\pi(50)=100\pi\) m.
Three laps are \(3(100\pi)=300\pi\) m.
18) The radius is \(8\). Whole pizza area is \(64\pi\).
One of \(8\) equal slices has area \(64\pi/8=8\pi\) square in.
19) Arc length: \(\frac{120}{360}\cdot2\pi(9)=6\pi\) cm.
Sector area: \(\frac{120}{360}\cdot\pi(9)^2=27\pi\) square cm.
20) \(18\pi=2\pi r\), so the original radius is \(9\).
Twice that radius is \(18\).
Area \(=\pi(18)^2=324\pi\).

More Practice

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