What is the measure of a heptagon?

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A heptagon is a polygon that has seven sides (Hepta- means seven). In the figure below are several types of heptagons.

Heptagon classifications

Like other polygons, a heptagon can be classified as regular or irregular. If all the sides and interior angles of a heptagon are equal, it is a regular heptagon. Otherwise it is an irregular heptagon.

Regular heptagon	Irregular heptagon
All sides and interior angles are equal	Not all sides and angles are equal

Heptagons and other polygons can also be classified as either convex or concave. If all interior angles of a heptagon are less than 180°, it is convex. If one or more interior angles is larger than 180°, it is concave. A regular heptagon is a convex heptagon. A concave heptagon is an irregular heptagon.

Convex heptagon	Concave heptagon
All interior angles <180°	One or more interior angles >180°

Diagonals of heptagon

A diagonal is a line segment joining two non-consecutive vertices. A total of fourteen distinct diagonals can be drawn for a heptagon. The following figure is an example.

There are 4 diagonals extending from each of the 7 vertices of the heptagon above creating a total of 14 diagonals.

Internal angles of a heptagon

The sum of the interior angles of a heptagon equals 900°.

As shown in the figure above, four diagonals can be drawn to divide the heptagon into five triangles. The blue lines above show just one way to divide the heptagon into triangles; there are others. The sum of interior angles of the five triangles equals the sum of interior angles of the heptagon. Since the sum of the interior angles of a triangle is 180°, the sum of the interior angles of the heptagon is 5 × 180° = 900°.

Regular heptagon

A regular heptagon is a heptagon in which all sides have equal length and all interior angles have equal measure.

Angles of a regular heptagon

Since each of the seven interior angles in a regular heptagon are equal in measure, each interior angle measures 900° ÷ 7 ≈ 128.57°, as shown below.

Each exterior angle of a regular heptagon has an equal measure that is approximately 51.43°.

Symmetry in a regular heptagon

A regular heptagon has 7 lines of symmetry and a rotational symmetry of order 7, meaning that it can be rotated in such a way that it will look the same as the original shape 7 times in 360°.

Lines of symmetry	Rotational symmetry
7 lines of symmetry	Seven 51.43-degree angles of rotation about the center

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home / geometry / shape / 3d

A three-dimensional space (3D) has three dimensions, such as length, width, and height (or depth). The term "3D" is commonly used to describe shapes and figures in geometry. We live in a 3D world, every object we touch, see, and use are 3D objects. The following is a few examples.

3D shapes and figures

While the dimensions of a 2D shape can be described with length and width, a 3D shape requires an additional dimension, often referred to as height or depth.

2D	3D
The dimensions of the 2D rectangle above are length and width.	The base of the 3D rectangular prism above is a rectangle that gives it two of its dimensions. The height of the prism is perpendicular to its base giving the prism its third dimension.

3D Coordinate geometry

Determining the position of a point in 3D is similar to determining the position of a point in 2D, except that there is a third axis, the z-axis, in addition to the x- and y-axes. All three axes are perpendicular to each other, as shown in the figure below.

While the x- and y-axes in 2D are conventionally the horizontal and vertical axes, respectively, in 3D their orientations can vary. As such, when determining which direction to move along any axis, the negative and positive direction must be indicated on the axes, as in the image above. A negative value indicates movement along an axis in the negative direction, and a positive value indicates movement in the positive direction. As such, the point (2, 3, 4) indicates movement in the positive direction along each axis in the coordinate space above.

See also geometric figures, 1D, 2D.

A heptagon is a 7-sided polygon with 7 interior angles that sum to 900°. The name heptagon derives from the Greek words hepta- for seven and gon- for sides. A heptagon is also called a 7-gon or septagon (septa- is Latin for seven).

Heptagon Shape

The heptagon shape is a plane or two-dimensional shape comprised of seven straight sides, seven interior angles, and seven vertices. A heptagon shape can be regular, irregular, concave, or convex.

Here are some additional properties of the heptagon shape:

• All heptagons have interior angles that sum to 900°
• All heptagons have exterior angles that sum to 360°
• All heptagons can be divided into five triangles
• All heptagons have 14 diagonals (line segments connecting vertices)

Heptagon sides must be straight and meet to form seven vertices that close in a space. The seven sides of a heptagon meet, but do not intersect, or cross over each other.

As with other 2d shapes, a heptagon's sides can be different lengths, which creates an irregular heptagon. Or the sides can be congruent to form a regular heptagon

A heptagon with sides that intersect is called a heptagram.

Heptagon Angles

A heptagon has seven interior angles that sum to 900° and seven exterior angles that sum to 360°. This is true for both regular and irregular heptagons.

In a regular heptagon, each interior angle is roughly 128.57°.

Below is the formula to find the measure of any interior angle of a regular polygon (n = number of sides):

We know that all heptagons (or septagons) have 7 sides, so we can plug that into our formula:

(180° × 7) - 360°7 =

1260° - 360°7 =

900°7 ≈ 128.5714°

Heptagon Diagonals

Heptagons have 14 diagonals. For convex heptagons, all diagonals will be inside the shape. For concave heptagons, at least one diagonal will be outside of the shape.

Regular Heptagon

Here is a picture of a regular heptagon. A regular heptagon has seven congruent sides, seven vertices, and seven congruent interior angles:

As indicated by the hash marks, the regular heptagon in the picture above has equal sides.

Convex Heptagon

A regular heptagon is always a convex heptagon. A convex heptagon has no interior angles greater than 179°:

Since no internal angle is greater than 179°, no diagonal can lie outside the polygon.

Irregular Heptagon

Here is an irregular heptagon, meaning its seven sides are not congruent and its seven interior angles are not identical:

Like other irregular polygons, irregular heptagons can be convex or concave like the heptagon image above.

Concave Heptagon

A concave heptagon has at least one interior angle greater than 180°, and it has at least one diagonal that lies outside the polygon:

Area Of A Heptagon

The area of a regular heptagon can be found using this formula:

This formula is approxmiatly equal to A = 3.643a2

In both formulas, a = side length.

Heptagon In Real Life

There are many examples of heptagon in real life, such as the two pictures below:

Like other geometric shapes, such as the octagon, hexagon, and quadrilateral, heptagonal figures can be found in man-made objects and in nature.

Heptagon Quiz

1. For any heptagon, what is the sum of its internal angles?
2. For any heptagon, how many vertices does it have?
3. For any heptagon, how many diagonals can you draw on it?
4. Can a heptagon have nine sides?
5. Below are some polygons. First, select all that are heptagons. Then, for each heptagon you select, determine if it is regular or irregular, and then whether it is concave or convex:

Please try the work before you take a peek at the answers!

1. The interior angles of a heptagon always add up to 900°.
2. All heptagons have seven vertices, just as they have seven sides and seven interior angles.
3. All heptagons will have 14 diagonals; if a diagonal lies outside the polygon, you know the heptagon is concave.
4. No, heptagons only have seven sides. A 9 sided polygon is called a nonagon.
5. Of the eight figures, only five are heptagons. Two are regular convex heptagons. Three are irregular concave heptagons. Bonus: one shape in the grid is a pentagon.

Next Lesson:

Decagon

Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.