# Angles, Polygons, and Area

Geometry is the study of the relationship between shapes, lines, points, and more.

A polygon is a flat figure that has at least three straight sides and angles. Some examples of polygons are triangles, squares, and rectangles.

The area of a polygon is the amount of space it is taking up. It can be useful to find the area of some of these shapes in real-life situations, such as if you are putting wallpaper on a rectangular wall or baking a pie and you are making the base dough.

Here is a cheat sheet of all the formulas, or equations, to find the area of the most common polygons.

Let’s try some examples and calculate the area of the following shapes!

Shown above is a triangle. We are given 8m as the height of the shape and the 6m as the base. As shown from our cheat sheet, the equation for finding the area of a triangle is ½ * base * height. We plug in our given values into our equation and the units are always squared (to the second power).

½ * base * height = ½ * 6m * 8m = 24 m^2

The answer is 24m^2! Let’s try finding the area of a circle.

The area of a circle is pi * r^2. In the picture above, the circle’s radius is 8ft, so we just have to plug our values in. Make sure to keep the units!

Area = pi * r^2 = pi * 8ft^2 = 64 * pi ft^2 (you do not have to multiply pi out) ≈ 201 ft^2

Now, for squares.

The area of a square is a length of one side squared or a^2. The side length in the square above is 5 cm so we just plug that into our equation.

Area = a^2 = 5 cm^2 = 25 cm^2

Let’s try one more shape, the rectangle, then we will move onto angles of a shape.

The area of a rectangle is width * length. The length of the rectangle is 7mm and the width is shown as 5mm. We can now calculate the area of the rectangle by just plugging in values.

Area = length * width = 7mm * 5mm = 35mm^2

Great! Now that we know how to calculate the areas of these shapes, we can now talk about the angles of a shape. The angle of a shape is formed by two lines of a shape and share a vertex (where the two lines intersect).

Each of the shaded parts are angles. Each vertex has an angle, so if a shape has N sides, then it has N angles. Regular polygons have all sides equal in length and each angle is equal in value. Shown below is another cheat sheet about angles and shapes. A quick way to calculate the value of each angle in a regular polygon is (180(n-2))/n, n being the number of sides.