# Adjacent Sides|Definition & Meaning

## Definition

**Adjacent sides** are any two sides of some **shape **(like a square or other polygons) that are **next to **each other and share a **common vertex**, forming a **non-zero** angle.

An adjacent side meets another side of the shape at some point. Adjacent means “**side-by-side or adjoining**,” and the **vertex** is the point at which two lines (or sides) join together. Therefore, when two sides **share a vertex**, they are **adjacent**.

It is possible in some complex shapes for a side to be adjacent to **more than just one** other side (if three or more sides join together at the same vertex).

## Understanding Adjacency Visually

Consider a triangle with vertices A, B, and C, as shown in the figure below.

The three sides of this triangle are **AB**, **BC**, and **CA**. Since sides **AB** and **BC** share the common vertex **B**, they are adjacent. Similarly, sides **BC** and **CA** share vertex **C**, and sides **CA** and **AB** share vertex **A**. Thus, there are always **three pairs of adjacent sides** in a triangle.

Now let us look at a pentagon (see Figure 2) with five vertices A, B, C, D, and E. Here, you have the sides AB, BC, CD, DE, and EA.

We can see that side **AB** is adjacent to sides **BC** (common vertex **B**) and **EA** (common vertex **A**). Similarly, we can see that side **BC** is next to **CD**, side **CD** is next to **DE**, and side **DE** is next to **EA**. They respectively share the vertices **C**, **D**, and **E**. Therefore, we have **five pairs of adjacent sides** here.

### Why Is Adjacency Important?

Adjacency is one of the critical **visual cues** for our brain to process the structured “oneness” of the shapes in our daily life. Like adjacent sides, we also have **adjacent vertices** and** angles**. These properties **simplify geometrical shapes** and allow us to analyze them properly.

For example, suppose you have **three matchsticks **arranged in the shape of a triangle. Now, imagine that you start taking the matchsticks **far away** from each other.

Eventually, the gap will be so large that it would be hard to think the matchsticks form a triangle! In this way, adjacency affects our perception of shapes.

## Adjacent Sides in 3D

The method of determining adjacent sides in 3D remains the same: identify the lines that **share a common vertex** – they are adjacent. A 3D shape **occupies** **space** (three dimensions: length, width, and depth).

Usually, adjacency in 3D is associated with the shape’s** faces**. Nonetheless, adjacent sides are equally valid.

For example, a **cube** has **six faces** (front, back, left, right, top, bottom). On its own, each face is a** square**. A square has **four vertices**, so each face has **four pairs of adjacent sides**.

Now consider one side on a single face of the cube. Since a side has two vertices, and each vertex connects up to three sides (including the side we are looking at) in a cube, each side is adjacent to **four** other sides! You can verify this from Figure 4 in Example 2.

## Opposite Sides

The **opposite side of** any given side is the one **farthest away **from it in the **opposite direction**. This is in contrast to adjacent sides. However, depending on the shape, a side can be opposite to another side or a vertex.

For example, if a polygon has an **even number **of sides, each side has an **opposite side**. On the other hand, if it has an **odd number **of sides, each side only has an **opposite vertex**.

## Examples of Finding Adjacent Sides

### Example 1

Find the total number of adjacent sides in a square with the diagonals drawn in and mention their shared vertices.

### Solution

With the diagonals drawn in, we now have eight sides: AB, BC, CD, DA, AE, CE, BE, and DE. The common vertices joining the adjacent sides are:

- A joins
**A**B,**A**E, and D**A** - B joins A
**B**,**B**E, and**B**C - C joins B
**C**,**C**E, and**C**D - D joins C
**D**,**D**E, and**D**A - E joins A
**E**, B**E**, C**E**, and D**E**.

Thus, each vertex has three adjacent sides, except for E, which has four. Adding them up, there are 2 x 4 + 6 = **14 pairs** of adjacent sides in total.

### Example 2

Brian is trying to figure out which sides of his friend’s 3-by-3 Rubik’s cube are adjacent. Extending the concept of adjacent sides to adjacent faces in 3D, try to solve the problem.

### Solution

We can see (from figure 4) a total of eight corner vertices and six faces on the Rubik’s cube. These corner vertices are A, B, C, D, E, F, G, and H. The six faces are then:

- ABFE:
**Front** - CDHG:
**Back** - BCGF:
**Right** - DAEH:
**Left** - EFGH:
**Top** - ABCD:
**Bottom**

Two faces are adjacent when they share at least one vertex. With this logic, we can see that the adjacent faces and the common vertices are:

- Front and Right (A
**BF**E and**B**CG**F)**: B + F - Front and Left (BF
**E**and D**AE**H): A + E - Front and Top (AB
**FE**and**EF**GH): E + F - Front and Bottom (
**AB**FE and**AB**CD): A + B - Back and Right (
**C**DH**G**and B**CG**F): C + G - Back and Left (C
**DH**G and**D**AE**H**): D + H - Back and Top (CD
**HG**and EF**GH**): G + H - Back and Bottom (
**CD**HG and AB**CD**): C + D - Right and Top (BC
**GF**and E**FG**H): F + G - Right and Bottom (
**BC**GF and A**BC**D): B + C - Left and Top (DA
**EH**and**E**FG**H**): E + H - Left and Bottom (
**DA**EH and**A**BC**D**): A + D

So there are a total of **12 adjacent face pairs** in a cube. We can do the same for each side of the cube as well!

*All mathematical drawings and images were created with GeoGebra.*