Circle Calculator

Circle calculator

The circle calculator is an online geometry calculator that you can use to find any of the following characteristics of a circle: radius, diameter, circumference, or area. The circle calculator takes one of the above characteristics as an input and calculates the other three characteristics.

The calculator uses the following notation:

• r – the radius of a circle,
• A – the area of a circle,
• C – the circumference of a circle,
• d – the diameter of a circle.

For the calculator to calculate the values listed above, it needs to use π. The value of π is assumed to be 3.1415926535898, but you can change this value in the corresponding field.

Directions for use

To use the calculator, choose the type of calculation from the drop-down list at the top of the calculator. The available types are:

1. Find A, C, and d | Given r;
2. Find C, r, and d | Given A;
3. Find A, r, and d | Given C;
4. Find A, C, and r | Given d.

Then input the known value – r, A, C, or d – into the corresponding field. In the following field, you can change the value of π (bear in mind that the default value used by the calculator is very accurate).

Note that the calculator also allows changing the units. The units do not influence the calculations; they are included for your convenience and to demonstrate the order of the resulting value. For example, the radius, r, can be measured in inches (in), which means the corresponding circle area, A, will be measured in square inches – in².

In the bottom drop-down list, you can select the number of significant values that are considered in the calculations. Once you have inputted everything, press "Calculate." The calculator will display the answers, solutions, and formulas used to find the answers.

Circle: definition and key formulas

In geometry, a circle is a two-dimensional curve, every point of which is at the same distance from a certain point – the center of the circle. The distance from the center of the circle to any point on the circular curve is called the radius. The line that connects two opposite points on the circumference and passes through the center of the circle is called the diameter. The diameter of a circle is always twice as long as the radius of the circle.

$$d = 2r$$

The circumference is the perimeter of the circle. You can use the following formula to find the circumference:

$$C = 2πr$$

Or, since the diameter is twice the radius:

$$C = πd$$

You can perform a backward calculation to find the radius from the circumference:

$$r = \frac{C}{2π}$$

Now let's look at how to find the area of a circle. You can calculate the area of a circle using any of the following formulas:

$$A = πr²$$

$$A = π \frac{d²}{4}$$

$$A = \frac{C²}{4π}$$

If the radius of a circle is known and the circle area is known, you can use the following formula:

$$r=\sqrt{\frac{A}{π}}$$

Calculation examples

Example 1

Find A, C, and d | Given r

Let's assume that the circle radius is known, and we need to find the other three values.

Given: r = 3 cm

Since the radius is known, we will choose the following type of calculation: Find A, C, and d | Given r. As the next step, we will input the value of "radius r" – 3. For convenience, we'll leave the default value alone and change the units to cm. We will use 3 significant figures to make the resulting answers less cumbersome.

Solution:

You can use the following formula to find the circle diameter:

$$d = 2r$$

Therefore, in our case:

$$d = 2r = 2 × 3 = 6$$

$$d = 6\ cm$$

To find the circumference, you can use the following formula:

$$C = 2πr$$

Therefore, in our case:

$$C = 2πr = 2 × π × 3$$

$$C = 6π$$

Considering that we want the answer to have only three significant figures, we get:

$$C = 18.8\ cm$$

To find the area, you can use the following formula:

$$A = πr²$$

Therefore, in our case:

$$A = πr² = π × 3²$$

Considering that we want the answer to have only three significant figures, we get:

$$A = 28.3\ cm²$$

Example 2

Find A, r, and d | Given C

Let's assume that the circumference is known, and we need to find the other three values.

Given: C = 10 in

Since the circumference is known, we will choose the following type of calculation: Find A, r, and d | Given C. Then we input the value of "circumference C" – 10. We will leave π at the default value and change Units to inches for convenience. Let's use 4 significant figures this time.

Solution:

To find the circle radius, you can use the following formula:

$$r = \frac{C}{2π}$$

Therefore, in our case:

$$r = \frac{C}{2π} = \frac{10}{2π}$$

Considering that we want the answer to have 4 significant figures, we get:

$$r = \frac{10}{6.2831853071796} = 1.592$$

$$r = 1.592\ in$$

To find the diameter, you can use the following formula:

$$d = \frac{C}{π}$$

Therefore, in our case:

$$d = \frac{C}{π} = \frac{10}{3.1415926535898}$$

Considering that we want the answer to have only four significant figures, we get:

$$d = 3.183\ in$$

To find the area, you can use the following formula:

$$A = \frac{C²}{4π}$$

or

$$A = πr²$$

Since we have already calculated the value of r.

Therefore, in our case:

$$A = πr² = π × 1.592² = 2.533 π$$

Considering that we want the answer to have only four significant figures, we get:

$$A = 7.958\ in²$$

Interesting facts about a circle

• The word "circle" comes from the Greek κίρκος/κύκλος (kirkos/kuklos), which means "ring" or "hoop."

• The invention of the circular wheel is considered one of the greatest inventions in the history of humanity.

• The circle has the shortest perimeter of all the geometrical shapes with the same area.

• The circle, along with the straight line, is the most widespread shape in all areas of human activity. In ancient times, circles and straight lines were often considered sacred shapes.

• Ancient scientists deemed only the circle and the straight line to be the perfect geometrical shapes. Therefore, in ancient geometry, they used only a pair of compasses and a ruler to construct other shapes and figures.

• The history of the circle is so ancient that it is impossible to say when people first identified this shape. The circle records exist in the oldest historical documents discovered, and people likely defined it much earlier.