Geometric Figure

Pronunciation: /ˌdʒi əˈmɛt rɪk ˈfɪg yər/ ?

A geometric figure is a set of one or more points in n-space.[1] The most basic geometric figure is a point. All other geometric figures are made from points. Lines and curves are continuous one-dimensional sets of points.

All geometric figures exist in a space. A space has certain properties. Geometric figures in a space must be consistent with the properties of that space.

Geometric Figures in n-Space

The only geometric figure that can exist in a space with 0 or more dimensions is a point. A geometric figure that can exist only in a space with one or more dimensions is a linear figure. Examples of linear objects are lines, rays, and line segments. A geometric figure that can exist only in a space with 2 or more dimensions is a planar figure. Examples of planar figures are circles, squares, and parabolas. A geometric figure that can exist only in a space with 3 or more dimensions is a solid. Examples of solids are cubes, spheres, and paraboloids.

Any geometric figure that can exist in a 1 dimensional space can also exist in a 2 or more dimensional space. More generally, a geometric figure that can exist in an n-space can exist in a space with more than n dimensions.

Properties of Geometric Figures

NameIllustrationDescription
Boundary point A four sided figure. A point inside the figure is labeled 'interior point'. A point on the edge of the figure is labeled 'boundary point'. A point outside the figure is labeled 'exterior point'. A point is a boundary point of a figure if every neighborhood of the figure contains points in the figure and points not in the figure. To visualize this, imagine a point in the figure that is not a boundary point. If one draws a small enough circle around that point, all points inside the circle will be part of the figure. The point can not be a boundary point. Only a point on the boundary fits this definition. Note that a boundary point may or may not be a part of the figure.
Boundary A four sided figure. The inside of the figure is labeled 'interior'. The edge of the figure is labeled 'boundary'. The outside of the figure is labeled 'exterior'. The boundary of a geometric figure is the set of all the boundary points of the figure.
Interior The interior of a geometric figure is all points that are part of the figure, but are not boundary points. Usually the interior of a geometric figure is what one thinks of when one thinks of an interior, but this is not necessarily so. Take, for example, a house. The interior of the house is considered to be the space inside the house. Now consider the outdoors (as opposed to the house and its interior). All parts of the outdoors are the interior of the outdoors and the interior of the house is the exterior of the outdoors.
Exterior The exterior of a geometric figure is all points that are not part of the figure, and are not boundary points.
Open A four-sided figure. The edges of the figure are solid lines. One edge is a dashed line. A geometric figure is open if and only if any part of the boundary of the figure is not a part of the figure. An open boundary is conventionally drawn with a dotted or dashed line.
Closed A four sided figure. The edges of the figure are all solid lines. A geometric figure is closed if and only if all of the boundary of the figure is part of the figure. A closed boundary is conventionally drawn with solid line.
Bounded A four sided figure. Five arrows start inside the figure and go to the edge of the figure. A geometric figure is bounded if, in any direction, there is always a last point that is part of the figure. To visualize this, imagine a ray drawn from a point on the figure. If all such rays encounter a last point that is part of the figure, the figure is bounded.
Unbounded A three sided figure where two of the sides go off into infinity. Five arrows start in the interior of the figure. Four of them go to the edge of the figure. The fifth goes off into infinity. A geometric figure is unbounded if, in any direction, there is no last point that is part of the figure. To visualize this, imagine a ray drawn from a point on the figure. If any such ray never encounters a last point that is part of the figure, the figure is unbounded.
Connected Two triangles. One of the triangles shares a vertex with the other triangle. A geometric figure is connected if, between all pairs of points that are part of the figure, there exists a path between the points that does not leave the figure.
Disconnected Two triangles. The triangles are separate and apart. A geometric figure is disconnected if, between any pair of points that are part of the figure, there exist no path between the points that does not leave the figure.
Dimension None The dimension of a geometric figure is hard to pin down without a discussion beyond the scope of this encyclopedia (definitely a college-level discussion). A brief example will have to do. A curve is a one-dimensional set of points. Yet a curve will not fit into a one dimensional space. A curve requires a space of at least 2 dimensions. A curve will fit into a 2 dimensional plane is called a planar curve. Consider also the surface of a sphere. The surface of a sphere has only 2 dimensions, yet it requires a 3-dimensional space to exist.
Table 1: Properties of Geometric Figures

Some Types of Geometric Figures

IllustrationNameDescription
PointA point labeled 'point'.An object of zero dimensions whose only property is location
LineA line going out of the image in both directions, a ray going out of the image in one direction. A line segment contained entirely within the image.An straight object of one dimension that goes on forever in both directions.
Line segmentA portion of a line with two endpoints.
RayA portion of a line with one endpoint.
CurveThree curves. One is an ellipse. It is a closed curve. One is a curve with one endpoint where the curve goes out of the image. The other is a curve with two endpoints.A curved line. A curve may have 0, 1 or 2 endpoints.
SquareA square. All four corners have angles showing that the corners are right angles. All for sides have a single hash mark through them showing that they are the same size.A four-sided figure with straight sides that meet at right angles.
CircleA circle. A line segment from the center of the circle to the edge is labeled 'radius'. The center of the circle is labeled 'center'.All points in a plane at the same distance from a center point.
ParabolaA parabola.The shape made when graphing a quadratic equation.
ParallelepipedA parallelepiped. All size sides are parallelograms. A solid whose sides are parallelograms.
SphereA sphere. A line from the center to the edge is labeled 'radius'.All points in 3 dimensions that are the same distance from a center.
HypersphereNoneAll points in 4 dimensions that are the same distance from a center.
Table 3: Some Types of Geometric Figures.

Cite this article as:


Geometric Figure. 2010-01-01. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/geometricfigure.html.

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2010-01-01: Added "More Information" and "References" (McAdams, David.)
2009-04-18: Initial version (McAdams, David.)

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