Distance Between 2 Points

Here is how to calculate the distance between two points when you know their coordinates:

Let us call the two points A and B

We can run lines down from A, and along from B, to make a Right Angled Triangle.

And with a little help from Pythagoras we know that:

a² + b² = c²

Now label the coordinates of points A and B.

x_A means the x-coordinate of point A
y_A means the y-coordinate of point A

The horizontal distance a is (x_A − x_B)

The vertical distance b is (y_A − y_B)

Now we can solve for c (the distance between the points):

Start with:c² = a² + b²

Put in the calculations for a and b:c² = (x_A − x_B)² + (y_A − y_B)²

And the final result:

Examples

Example 1

Fill in the values:

Example 2

It doesn't matter what order the points are in, because squaring removes any negatives:

Fill in the values:

Introduction to Coordinate Geometry

A system of geometry where the position of points on the plane is described using an ordered pair of numbers.

Recall that a plane is a flat surface that goes on forever in both directions. If we were to place a point on the plane, coordinate geometry gives us a way to describe exactly where it is by using two numbers.

What are coordinates?

To introduce the idea, consider the grid on the right. The columns of the grid are lettered A,B,C etc. The rows are numbered 1,2,3 etc from the top. We can see that the X is in box D3; that is, column D, row 3.

D and 3 are called the coordinates of the box. It has two parts: the row and the column. There are many boxes in each row and many boxes in each column. But by having both we can find one single box, where the row and column intersect.

The Coordinate Plane

In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales - one running across the plane called the "x axis" and another a right angles to it called the y axis. (These can be thought of as similar to the column and row in the paragraph above.) The point where the axes cross is called the origin and is where both x and y are zero.

On the x-axis, values to the right are positive and those to the left are negative.
On the y-axis, values above the origin are positive and those below are negative.

A point's location on the plane is given by two numbers,the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15. These are the coordinates of the point A, sometimes referred to as its "rectangular coordinates". Note that the order is important; the x coordinate is always the first one of the pair.

For a more in-depth explanation of the coordinate plane see The Coordinate Plane.
For more on the coordinates of a point see Coordinates of a Point

Things you can do in Coordinate Geometry

If you know the coordinates of a group of points you can:

• Determine the distance between them
• Find the midpoint, slope and equation of a line segment
• Determine if lines are parallel or perpendicular
• Find the area and perimeter of a polygon defined by the points
• Transform a shape by moving, rotating and reflecting it.
• Define the equations of curves, circles and ellipses.

Metric Length

We can measure how long things are, or how tall, or how far apart they are. Those are are all examples of length measurements.

Example: This fork is 20 centimeters long

These are the most common measurements:

• Millimeters
• Centimeters
• Meters
• Kilometers

Small units of length are called millimeters.

A millimeter is about the thickness of a plastic id card (or credit card).

Or about the thickness of 10 sheets of paper on top of each other.

This is a very small measurement!

When we have 10 millimeters, it can be called a centimeter.

1 centimeter = 10 millimeters

A fingernail is about one centimeter wide.

Two tape measures, one in mm, the other in cm

We can use millimeters or centimeters to measure how tall we are, or how wide a table is, but to measure the length of football field it is better to use meters.

A meter is equal to 100 centimeters.

1 meter = 100 centimeters

The length of this guitar is about 1 meter

Meters can be used to measure the length of a house, or the size of a playground.

And because a centimeter is 10 millimeters:

1 meter = 1000 millimeters

The back of this kitchen chair is 1 meter

A kilometer is equal to 1000 meters.

When we need to get from one place to another, we measure the distance using kilometers.

The distance from one city to another or how far a plane travels can be measured using kilometers.

Final thoughts about measuring length:

1 centimeter = 10 millimeters

1 meter = 100 centimeters

1 kilometer = 1000 meters

Measurement Activities

Measuring is an important practical skill that is frequently used to carry out real life activities such as buying groceries or painting a house. Teaching your child measurement skills doesn’t have to revolve around textbooks or solving problems on paper.

Make measuring fun for your child with the free printable activities and games here.

Free Measuring Activities

• Making Lemonade with Mom
• Homemade Ruler
• Footprints on Papers
• Card Measurement
• Box Castle
• Measuring a Tree
• Guess, Measure, Weigh

More about Measurement Activities

Kids learn more by doing rather than listening or reading and this is truer for measurement skills than anything else. You can pique your child’s interest by engaging him with measuring real objects making measurement a part of his daily routine. Younger children, in particular, enjoy simple activities such as arranging objects by size, while older kids are able to work with more difficult concepts such as area and perimeter.

Here are a number of hands-on activities that teach your child to measure in a fun and engaging manner. Not only are they interactive and interesting, they also help your child apply the concept of measurement to real-world contexts.

And don’t forget to check out our free math activities for kids of all ages.