Formulas and Practice for Finding the Area of 2D Shapes

Welcome to a guide about the area of 2D shapes, one of the most fundamental parts of geometry! There are many practical applications to this concept in the real world, including engineering, architecture, and even business, so students—take careful notes! 📝

Throughout this article, we will discuss calculating the area for a rectangle, triangle, trapezoid, parallelogram, and circle. 📐

Image Courtesy of Wikimedia

Area of a Rectangle

The basic formula for calculating area is multiplying the length of the object by the width of the object. Therefore, the formula for the area of a rectangle is the length times width. 💡

Example Problem 1

If the length of a rectangle is 5 meters and the width of it is 8 meters, what is the area of the rectangle?

• Formula: length x width

• Input variables: 5 x 8 

• Answer: 40 m squared

Example Problem 2

If a rectangular-sized field that needs to be watered is 100 yards wide and 50 yards long, how much area does the water need to cover?

• Formula: length x width

• Input variables: 50 x 100

• Answer: 5000 yards squared

Area of a Triangle

A triangle is essentially a rectangle or square that has been sliced in half diagonally. This means that we can use the basic area formula of a rectangle and then divide by two to get the area of a triangle! ✂️

Example Problem 1:

If a triangle is fifteen inches long and 10 inches wide, what is the area of the triangle?

• Formula: (length x width) / 2

• Input variables: (15 x 10) / 2

• Answer: 75 inches squared

Example Problem 2:

If Rachel draws a triangle that is 8 centimeters wide and 5 centimeters wide, how much area does she need to cover to color her triangle?

• Formula: (length x width) / 2

• Input variables: (8 x 5) / 2

• Answer:  20 cm squared

Area of a Parallelogram

A parallelogram is essentially a rectangle in a different formation. Thus, the formula is also the same. The animation below provides a great visualization! 👍

Image Courtesy of GeoGebra

Example Problem 1:

If a parallelogram is 18 meters long and 10 meters wide, what is the area of the shape?

• Formula: (length x width) 

• Input variables: (18 x 10) 

• Answer: 180 m squared

Example Problem 2:

If Michael constructs a parallelogram from pipe cleaners that is 20 centimeters wide and 8 centimeters wide, how much area does the parallelogram cover?

• Formula: (length x width) 

• Input variables: (20 x 8) 

• Answer:  160 cm squared

Area of a Trapezoid

The area for a trapezoid is found by dividing the sum of the parallel sides of a trapezoid by two. Then, multiply it by the height. 🧠

Example Problem 1:

Dua draws a trapezoid for art class. One parallel side is 5 cm. The other parallel side is 9 cm. The height is 10 cm. What area does she need to cover using her blue marker?

• Formula: (parallel side 1 + parallel side 2 / 2) X height

• Input variables: (5 + 9 / 2) * 10 = 7 * 10

• Answer: 70 cm squared

Example Problem 2:

The perimeter of the trapezoidal field is 20 yards. The two non-parallel sides are each 5 yards. The height of the trapezoid is 10 yards. 

• Formula: (parallel side 1 + parallel side 2 / 2) X height

• Input variables: (10/2) * 10

• We know that the sum of the parallel sides is 10yards  because the other two sides are 5 yards each. If we subtract five yards two times from the perimeter, we are left with 10 yards. These must be for the parallel sides of the trapezoid.

• Answer: 50 yards squared

Area for a Circle

The circle’s area formula is a bit different than the other shapes we covered. We use the irrational number pi (π) and multiply it by the radius (r) squared. The radius is half of the length of the circle, so the diameter is the full length of the circle. 

Example Problem 1:

The radius of the circular cookie mold is 35 meters. What is the area?

• Formula: (π x r^2)

• Input variables: (π x 35^2) 

• Answer: 3848.45 m squared

Example Problem 2:

The diameter of the circular rug is 10 meters. What is the area?

• Formula: (π x r^2)

• Input variables: (π x 5^2)

  • We know that the radius is 5 because the radius equals half the diameter.

• Answer: 78.54 m squared

Area of a Sector of a Circle

Image Courtesy of Math Stack Exchange

First, let’s define what a sector is—a portion of a circle with two radii and the corresponding arc! To find the area of a sector of a circle, multiply the previous section formula by the percent of the circle that you have to find the area for. 🧐

Example Problem :

You cut out a wooden circle for an engineering class. It is 180 degrees and has a radius of 10 centimeters.

• Formula: ( degrees/360 x πr^2)

• Input variables: (180/360 x π10^2) = (0.5 x 100π) 

• Answer: 157 cm squared

Conclusion

Congratulations—you’ve learned all there is to know about the area of 2D shapes. To practice or explore related concepts, visit Fiveable for more resources. Now go conquer your math class! 🤓