How to Find the Area and Perimeter

Hey guys! Welcome to this video on finding the area and perimeter of a shape.

The area and perimeter help us to measure two-dimensional shapes.

The area measures the surface of a shape, and perimeter measures the distance around the outside of a shape.

We’ll take a look at how to calculate each, but let’s start with perimeter.

Calculating Perimeter

To calculate the perimeter of any polygon, you add up the length of all the sides. For reference, a polygon is any two-dimensional closed shape that’s composed of straight lines.

Perimeter of a Rectangle

Let’s use a rectangular swimming pool as an example.

We want to add a safety fence along the outside of the pool. In order to figure out how much material is needed, we first need to figure out the perimeter of the pool. The length of the pool is 15 ft, and the width is 7 ft.

Because it’s a rectangle, we know that the opposite sides are congruent to each other.

This is all the information we need. Let’s add up all of the sides!

Assuming 15 is equal to \(a\), and 7 is equal to \(b\), we can use the following formula: \(\text{Perimeter} = a + a + b + b\).

Now let’s plug in our numbers. \(P=15\text{ ft}+15\text{ ft}+7\text{ ft}+7\text{ ft}\). Added together, we get 44 feet.

Alternatively, since we know that there are four sides and that there are two sets of identical sides, we can simplify our formula to \(P=2a+2b)\). Let’s give it a whirl—I bet we get the same result.

Perimeter of a Square

We can apply this same sort of principle to finding the perimeter of a square.

A square has four equal sides, so we can say that the perimeter of a square is \(4\times a\).

This principle also applies with an equilateral triangle. Because all three sides are the same, \(P=3\times a\). An isosceles triangle has two sides that are the same, so the perimeter is \(2a+b\). I think you get the point.

Now, the only shape whose perimeter may seem a little less obvious to find is our friend the circle. You may have heard of the term circumference. Circumference is the measurement of the distance around a circle, which is similar to the perimeter of a polygon.

Because there are no straight sides on a circle, the circumference can be calculated by multiplying the diameter of the circle times pi (\(C=\pi d\)) or by multiplying the radius times 2 times pi (\(C=2\pi r\)).

Okay, we’ve covered perimeter, now let’s move on to area.

How to Find Area

When we were looking at the perimeter of different shapes, we were able to simplify the formulas. This gave us different formulas for different shapes, but each one followed the same principles.

Similarly, when finding the area of an object, the formula will be different for different shapes, but they all serve the same purpose: calculating the amount of space a shape occupies.

So, let’s take a look at different shapes and their area formulas.

Area Formulas

A square, a rectangle, and a parallelogram all share a formula, which is \(A+bh\), commonly referred to as “length times width.”

The area of a triangle is \(A=\frac{1}{2}bh\).

The area for a trapezoid is \(A=\frac{1}{2}(b_1+b_2)h\).

The area of a circle is \(A=\pi r^2\).

Now let’s look at some examples of how to solve for the area of an triangle and the area of a trapezoid.

Area of a Triangle Example

Given that a triangle has a base of 6 centimeters and a height of 4 centimeters, calculate the area by substituting the values into the triangle area formula (\(A=\frac{1}{2}bh\). It’s important to know that the height of a triangle will always make a right angle with the base.

 
Start by multiplying 6 times 4 to get a product of 24.

 
Then, multiply 1/2 times 24 (the same as dividing by 2), which gives us a solution of 12 cm².

Area of a Trapezoid Example

Now, let’s say you have a trapezoid with a height equal to 7 inches, base 1 equal to 8 inches, and base 2 equal to 6 inches.

Given this, we can substitute our numbers into the formula for the area of a trapezoid:

 
Start by finding the sum of 8 and 6.

 
Then, multiply 14 by 7 to get a product of 98.

 
Finally, you can multiply 98 by 1/2 or divide 98 by 2 to get a solution of 42 in².

 
That’s it for this review. Thanks for watching, and happy studying!

Frequently Asked Questions

Area and Perimeter Practice Problems