---
title: "Polygon | Microsoft Docs"
ms.date: "03/06/2017"
ms.prod: "sql-server-2014"
ms.reviewer: ""
ms.technology: 
ms.topic: conceptual
helpviewer_keywords: 
  - "geometry subtypes [SQL Server]"
  - "Polygon geometry subtype [SQL Server]"
ms.assetid: b6a21c3c-fdb8-4187-8229-1c488454fdfb
author: MladjoA
ms.author: mlandzic
manager: craigg
---
# Polygon
  A `Polygon` is a two-dimensional surface stored as a sequence of points defining an exterior bounding ring and zero or more interior rings.

## Polygon instances
 A `Polygon` instance can be formed from a ring that has at least three distinct points. A `Polygon` instance can also be empty.

 The exterior and any interior rings of a `Polygon` define its boundary. The space within the rings defines the interior of the `Polygon`.

 The illustration below shows examples of `Polygon` instances.

 ![Examples of geometry Polygon instances](../../database-engine/media/polygon.gif "Examples of geometry Polygon instances")

 As shown in the illustration:

1.  Figure 1 is a `Polygon` instance whose boundary is defined by an exterior ring.

2.  Figure 2 is a `Polygon` instance whose boundary is defined by an exterior ring and two interior rings. The area inside the interior rings is part of the exterior of the `Polygon` instance.

3.  Figure 3 is a valid `Polygon` instance because its interior rings intersect at a single tangent point.

### Accepted instances
 Accepted `Polygon` instances are instances that can be stored in a `geometry` or `geography` variable without throwing an exception. The following are accepted `Polygon` instances:

-   An Empty `Polygon` instance

-   A `Polygon` instance that has an acceptable exterior ring and zero or more acceptable interior rings

 The following criteria are needed for a ring to be acceptable.

-   The `LineString` instance must be accepted.

-   The `LineString` instance must have at least four points.

-   The starting and ending points of the `LineString` instance must be the same.

 The following example shows accepted `Polygon` instances.

```
DECLARE @g1 geometry = 'POLYGON EMPTY';
DECLARE @g2 geometry = 'POLYGON((1 1, 3 3, 3 1, 1 1))';
DECLARE @g3 geometry = 'POLYGON((-5 -5, -5 5, 5 5, 5 -5, -5 -5),(0 0, 3 0, 3 3, 0 3, 0 0))';
DECLARE @g4 geometry = 'POLYGON((-5 -5, -5 5, 5 5, 5 -5, -5 -5),(3 0, 6 0, 6 3, 3 3, 3 0))';
DECLARE @g5 geometry = 'POLYGON((1 1, 1 1, 1 1, 1 1))';
```

 As `@g4` and `@g5` show an accepted `Polygon` instance may not be a valid `Polygon` instance. `@g5` also shows that a Polygon instance needs to only contain a ring with any four points to be accepted.

 The following examples throw a `System.FormatException` because the `Polygon` instances are not accepted.

```
DECLARE @g1 geometry = 'POLYGON((1 1, 3 3, 1 1))';
DECLARE @g2 geometry = 'POLYGON((1 1, 3 3, 3 1, 1 5))';
```

 `@g1` is not accepted because the `LineString` instance for the exterior ring does not contain enough points. `@g2` is not accepted because the starting point of the exterior ring `LineString` instance is not the same as the ending point. The following example has an acceptable exterior ring, but the interior ring is not acceptable. This also throws a `System.FormatException`.

```
DECLARE @g geometry = 'POLYGON((-5 -5, -5 5, 5 5, 5 -5, -5 -5),(0 0, 3 0, 0 0))';
```

### Valid instances
 The interior rings of a `Polygon` can touch both themselves and each other at single tangent points, but if the interior rings of a `Polygon` cross, the instance is not valid.

 The following example shows valid `Polygon` instances.

```
DECLARE @g1 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20))';
DECLARE @g2 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0))';
DECLARE @g3 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (-10 0, 0 10, -5 -10, -10 0))';
SELECT @g1.STIsValid(), @g2.STIsValid(), @g3.STIsValid();
```

 `@g3` is valid because the two interior rings touch at a single point and do not cross each other. The following example shows `Polygon` instances that are not valid.

```
DECLARE @g1 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (20 0, 0 10, 0 -20, 20 0))';
DECLARE @g2 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (5 0, 1 5, 1 -5, 5 0))';
DECLARE @g3 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (-10 0, 0 10, 0 -10, -10 0))';
DECLARE @g4 geometry = 'POLYGON((-20 -20, -20 20, 20 20, 20 -20, -20 -20), (10 0, 0 10, 0 -10, 10 0), (-10 0, 1 5, 0 -10, -10 0))';
DECLARE @g5 geometry = 'POLYGON((10 0, 0 10, 0 -10, 10 0), (-20 -20, -20 20, 20 20, 20 -20, -20 -20) )';
DECLARE @g6 geometry = 'POLYGON((1 1, 1 1, 1 1, 1 1))';
SELECT @g1.STIsValid(), @g2.STIsValid(), @g3.STIsValid(), @g4.STIsValid(), @g5.STIsValid(), @g6.STIsValid();
```

 `@g1` is not valid because the inner ring touches the exterior ring in two places. `@g2` is not valid because the second inner ring in within the interior of the first inner ring. `@g3` is not valid because the two inner rings touch at multiple consecutive points. `@g4` is not valid because the interiors of the two inner rings overlap. `@g5` is not valid because the exterior ring is not the first ring. `@g6` is not valid because the ring does not have at least three distinct points.

## Examples
 The following example creates a simple `geometry``Polygon` instance with a hole and SRID 10.

```
DECLARE @g geometry;
SET @g = geometry::STPolyFromText('POLYGON((0 0, 0 3, 3 3, 3 0, 0 0), (1 1, 1 2, 2 1, 1 1))', 10);
```

 Aninstance that is not valid may be entered and converted to a valid `geometry` instance. In the following example of a `Polygon`, the interior and exterior rings overlap and the instance is not valid.

```
DECLARE @g geometry;
SET @g = geometry::Parse('POLYGON((1 0, 0 1, 1 2, 2 1, 1 0), (2 0, 1 1, 2 2, 3 1, 2 0))');
```

 In the following example, the invalid instance is made valid with `MakeValid()`.

```
SET @g = @g.MakeValid();
SELECT @g.ToString();
```

 The `geometry` instance returned from the above example is a `MultiPolygon`.

```
MULTIPOLYGON (((2 0, 3 1, 2 2, 1.5 1.5, 2 1, 1.5 0.5, 2 0)), ((1 0, 1.5 0.5, 1 1, 1.5 1.5, 1 2, 0 1, 1 0)))
```

 Here is another example of converting an invalid instance to a valid geometry instance. In the following example the `Polygon` instance has been created using three points that are exactly the same:

```sql
DECLARE @g geometry
SET @g = geometry::Parse('POLYGON((1 3, 1 3, 1 3, 1 3))');
SET @g = @g.MakeValid();
SELECT @g.ToString()
```

 The geometry instance returned above is a `Point(1 3)`.  If the `Polygon` given is `POLYGON((1 3, 1 5, 1 3, 1 3))` then `MakeValid()` would return `LINESTRING(1 3, 1 5)`.

## See Also
 [STArea (geometry Data Type)](/sql/t-sql/spatial-geometry/starea-geometry-data-type) 
 [STExteriorRing (geometry Data Type)](/sql/t-sql/spatial-geometry/stexteriorring-geometry-data-type) 
 [STNumInteriorRing (geometry Data Type)](/sql/t-sql/spatial-geometry/stnuminteriorring-geometry-data-type) 
 [STInteriorRingN (geometry Data Type)](/sql/t-sql/spatial-geometry/stinteriorringn-geometry-data-type) 
 [STCentroid (geometry Data Type)](/sql/t-sql/spatial-geometry/stcentroid-geometry-data-type) 
 [STPointOnSurface (geometry Data Type)](/sql/t-sql/spatial-geometry/stpointonsurface-geometry-data-type) 
 [MultiPolygon](../spatial/polygon.md) 
 [Spatial Data (SQL Server)](../spatial/spatial-data-sql-server.md) 
 [STIsValid (geography Data Type)](/sql/t-sql/spatial-geography/stisvalid-geography-data-type) 
 [STIsValid (geometry Data Type)](/sql/t-sql/spatial-geometry/stisvalid-geometry-data-type)