Definition of a Diameter
In maths, the diameter is a straight line from one circle’s edge to the opposite edge of the same circle. So this line must pass through the circle’s center to be called a diameter.
Another way of putting it: the diameter is the longest distance measured across a circle.
Definition of a Radius
There is also another circle component closely related to the definition of the diameter. That component is the radius. The radius is a line drawn from the circle’s edge to its center.
The diameter and radius are essential values required to complete circle circumference and area calculations.
Why are Diameters Important?
Diameters are briefly introduced in KS2 lessons but are more fully explored in KS3 maths classes and higher stages. In these classes, the objective is to present the formulas for the area and circumference of a circle and demonstrate how students may use diameters to solve the equations.
In the real world, there are numerous examples of circles. Therefore, knowing how to find the area and circumference of these circles is often critical.
Real-World Applications of Diameters
Tires on a Car
Tires come in multiple sizes. So, when we need to replace the tires on a car, it is essential to know which size tire will fit to avoid costly mistakes. The diameter of the wheel is a crucial factor when it comes to selecting the right tire.
For example, take the tire size P225/65R17. But what does this mean? Along with telling us the tire width and aspect ratio, these numbers on the sidewall tell us the wheel size the tire is intended to fit. The last number tells us that the tire fits a wheel with a 17-inch diameter. If the wheel diameter of a car is 15 inches, this tire would not be suitable.
Plates in the Kitchen
A dinner plate’s diameter is also a tremendous real-world application. One consideration that may be overlooked when purchasing a new dish is whether the container can be easily stored at home. Some kitchen cabinets are not deep enough to stack plates with a diameter over a specific size. A plate’s diameter can also impact whether or not the container can be placed in the dishwasher without obstructing the spray arms.
Arts and Crafts
Arts and crafts also have a use for diameters. For example, creating a circular tabletop mosaic. One of the first steps in this project is to know the surface area of the tabletop. This information helps to ensure you have adequate materials for your project. Knowledge of a circular table’s diameter or width is necessary to calculate the area. Once the area is known, it’s easier to determine the number of materials required to cover the table.
Bringing Diameters Back to Maths
Diameters are a critical value necessary to make circle calculations.
Unlike shapes like triangles, rectangles, and squares, it’s not easy to measure the length of a circle’s perimeter, known as circumference, using a ruler. Nor is it possible to calculate the area based on the measurements of the sides. So instead, we use the diameter size to calculate the circumference and location of a circle.
There are two additional points about diameters that are good to remember:
- Diameters are the most extended chord in a circle. The straight line from one edge of the circle to the other must pass through the center. If the line doesn’t pass through the center but still has endpoints that lie on the circular arc, that line is called a chord. All diameters are chords, but not all chords are diameters.
- The radius is the distance from the circle’s edge to the center. The radius multiplied by 2 equals the diameter of the circle.
Diameters and Calculating the Circumference and Area
Difference Between the Radius and Diameter of a Circle
Both diameter and radius are critical when it comes to calculating the circumference and area of a circle. And we have already defined both of these segments. But let’s look at the relationship between these two lines closely.
As the radius measures the distance from the edge of the circle to its center, we can confidently state that the radius is half the value of the diameter. Alternatively, we can say that the diameter equals two times the radius. The following equation represents this statement:
diameter = 2 × radius
The formula is also written as:
d = 2r
where d is the diameter and r is the radius
Let’s put this formula into practice.
| Example 1:
A circle has a diameter of 6 inches. First, find the radius of the circle. As the diameter is twice that of the radius, we can solve the problem using our maths skills with the following equation: d = 2r 6 = 2 × r 6 ÷ 2 = (2 × r) ÷ 2 3 = r The circle has a radius of 3 inches. Example 2: A circle has a radius of 12 cm. Find the circle’s diameter. Again, we use our knowledge of the diameter and radius relationship to solve the problem: d = 2r d = 2 × 12 d = 24 cm The diameter of the circle is 24 cm. |
Introducing Pi (π)
We also use the number pi, pronounced as pie and written as π, when finding the circumference and area of a circle. There is a π button on most calculators, or you can use the approximate value of π, which is 3.14.
Pi, or π, is what mathematicians call an endless number. These numbers have a value that does not change; it is the same in every equation and circumstance. However, the number π is also irrational, meaning that the number does not end or have a repeating pattern. We, therefore, approximate π to 3.14 as a decimal.
There is also an annual holiday to celebrate π, called Pi Day. Pi Day is celebrated on the 14th of March. This is because the approximate value of π matches how some countries, like the United States, write the calendar date of this holiday: month. Day = 3.14. The holiday gives people the opportunity to focus on all things pi, whether that be the maths or edible variety.
Let’s Discuss the Circumference of a Circle
The circumference is the measured distance around a circle, similar to a perimeter. If you use a piece of string to trace around the edge of the circle and then measure the string length on a ruler, you would measure the circumference. Fortunately, there’s a maths formula using the diameter value that helps you quickly find the circumference of a circle, saving you time (and string).
To find the circumference of a circle, we use the following equation:
Circumference of a circle = π × diameter of the circle
In more simple maths terms, this equation is written as:
C = π × d or C = πd
where C is the circumference and d is the diameter
As we learned earlier that the diameter is twice the circle’s radius, you can use either of these values to find the circle’s circumference. The only difference is that the radius must be multiplied by two before calculating the circumference:
C = πd
or
C = 2×π×r or C = 2πr
where C is the circumference and r is the radius
Therefore, C = πd is the same as C = 2πr.
| Example 1:
A circle has a diameter of 18 cm. Find the circumference of the circle and round it to two decimal places. In this example, we know that the diameter is 18 cm. Therefore, the equation is written as follows: C = πd C = π × 18 C = 56.548667764…cm The circumference of the circle is 56.55 cm. Example 2: A circle has a radius of 5 cm. Find the circumference of the circle and round it to two decimal places. In this equation, we know that the radius is half the diameter. Therefore, the radius must be multiplied by 2 to solve the problem. C = 2πr C = 2 × π × 5 C = π × (5 × 2) C = π × 10 C = 31.4159265…cm The circumference of the circle is 31.42 cm. |
Let’s Look at the Area of a Circle
It’s the radius and not the diameter that is the critical value when it comes to finding the area of a circle. However, it’s still important to understand the relationship between diameters and radiuses if only the diameter value is given.
To find the area of a circle, we use the following equation:
Size of a circle = π × radius of the circle × radius of the circle
As we’re multiplying the radius by itself, we say that we are squaring the radius. Therefore, the above equation can be written in the following way:
Area of a circle = π × (radius of the circle)²
In maths notations, this equation is written as:
A = π × r² or A = πr²
where A is the area and r is the radius
| Example 1:
A circle has a radius of 7 cm. Find the area of the circle and round to two decimal places. In this example, we are given the radius, or r, as 7 cm. We then insert the values into the equation as follows: A = πr² A = π × (7)² is the same as A = π × (7×7) A = π × 49 A = 153.938040…cm² The area of the circle is 153.94 cm². Example 2: A circle has a diameter of 18m. Find the area of the circle and round to two decimal places. In this example, we are given a diameter of 18m. However, we need the radius to find the circle’s area. As the radius is half the diameter, we divide the diameter by 2. radius = diameter ÷ 2 radius = 18 ÷ 2 radius = 9 We can now find the area as in the previous example: A = πr² A = π × (9 × 9) A = π × 81 A = 254.4690049…m² The area of the circle is 254.47 m². |
Top Tip: The area of any shape is given in square units, so make sure to add the square unit notation. This can be cm², ft², in², or others.
