Area and Perimeter of an L-Shape

TR Smith is a product designer and former teacher who uses math in her work every day.

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There are several equivalent methods to find the perimeter and area of an L-shape.

An L-shape (also called an ell or ell-shape) is a common layout for rooms and outdoor areas. If you want to calculate how much trim you need to cover every side of an L-shape, you need to know the perimeter. To calculate the amount of material needed to cover the surface, you need to know the area. There are several ways to figure the perimeter and area of an L-shaped room; all produce the correct answer, so it is only a matter of preference which one you use.

Perimeter of an L-Shape

There are two ways to compute the length of the boundary of an L-shape. The first is to simply add up the lengths of all six of its sides. For example, in the image below, the lengths of the sides of the L are 15, 12, 4, 7, 11, and 5, going counter-clockwise from the top edge. Therefore its perimeter is 15 + 12 + 4 + 7 + 11 + 5 = 54.

The second way to compute is the perimeter of an L-shape is to add the lengths of the two sides that form the outside corner of the L-shape, and then double it. In other words, compute 2(A+B), where A is the total length of the shape and W is the total width. Does this look like the formula for the perimeter of a rectangle? It should, because both perimeter formulas are the same!

Proof: If you look at the perpendicular sides that form the inside corner, you can rotate them 180 degrees so that they complete the border of a rectangle. Since nothing was added or subtracted in this process, the perimeter of the L-shape equals the perimeter of the new rectangle. The new rectangles sides are A, B, A, and B, so the perimeter is 2(A+B).

In this example, the perimeter is 2(12+15) = 2*27 = 54, which matches the previous answer.

Area of an L-Shape

There are two ways to find the area of an L-shape, one is to partition it into two smaller rectangles and add their areas, the other is to look at it as the difference in areas between a larger and smaller rectangle.

For the first method, there are two ways you can section it off into two rectangles. The figure below shows one way. As you can see, one of the smaller rectangles has dimensions 4-by-12, and the other has dimensions 5-by-11. Thus, the total area of the figure is

4*12 + 5*11 = 48 + 55 = 103.

The second way to find the area of an L-shape is to look at it as the space between a smaller rectangle and a larger rectangle. If you look at the same example below, the larger rectangle has dimensions 12-by-15. The smaller rectangle has dimensions 7-by-11. Subtracting their areas gives you the area of the L-shape:

12*15 - 7*11 = 180 - 77 = 103.

This matches the answer obtained with the previous method.

Other Geometry Tutorials

L-Shape Calculator: Computes the area and perimeter of any L-shape when you input its measurements.

Inscribed Circles/Squares: How to solve geometry problems involving circles inscribed in squares.

Annulus Tutorial: How to find the area of an annulus with three different formulas.

Pyramidal Frustum: How to find the volume of a truncated square pyramid, aka square pyramidal frustum.

Barrel Volume: How to estimate the volume of a barrel-shaped container with three different formulas.

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