Hexagon Shape
- Introduction
- What is a Hexagon?
- Types of Hexagons
- Understanding Hexagon Sides
- Understanding Angles in a Regular Hexagon
- Understanding Angles in an Irregular Hexagon
- Perimeter of a Hexagon
- Properties of a Hexagon
- Diagonals of a Hexagon
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
In geometry, a hexagon is a polygon with six sides and six angles. It's a shape that's commonly found in nature, two such instances can be the shape of the cells in a beehive or the shape of snowflakes. Additionally, hexagons are quite versatile in various geometric constructions and patterns. When you look at a stop sign, you see a regular hexagon, where all six sides are of equal length and all six angles are equal. The introduction of hexagons in geometry allows us to explore the properties of six-sided shapes helping us understand more complex geometric concepts.
What is a Hexagon?
A hexagon is a geometric shape with six straight sides and six angles. It resembles a stop sign, with each side connecting to the next at a corner. This shape is not hard to find in nature, like in honeycomb cells or the structure of certain crystals. When all its sides are of equal length and all its angles are equal, it's called a regular hexagon. Regular hexagons are special in geometry because of their symmetry and the various ways they can be used in design and construction, from building blocks to intricate patterns.
Types of Hexagons
`1`. Regular Hexagon: A regular hexagon is a hexagon with all sides of equal length and all angles of equal measure. It has six symmetry lines, splitting it into two equal halves. Regular hexagons are often seen in real life like the outer shape of nuts and bolts, and are commonly used in architecture and design due to their uniformity and appealing symmetry.
`2`. Irregular Hexagon: An irregular hexagon is a hexagon with sides of varying lengths and/or angles of different measures. Unlike the regular hexagon, it lacks symmetry and uniformity. Irregular hexagons can have a wide range of shapes and configurations, making them versatile in various contexts but also challenging to work with geometrically.
`3`. Convex Hexagon: A convex hexagon is a hexagon where all its interior angles are less than `180°`. This means no line segment connecting two non-adjacent vertices of the hexagon lies entirely outside the shape. Convex hexagons have no "dents" or inward-pointing angles in their outlines.
`4`. Concave Hexagon: A concave hexagon is a hexagon with at least one interior angle greater than `180°`. This results in at least one "dent" or inward-pointing angle in its outline. Concave hexagons can have irregular shapes with portions of the perimeter folded inward.
`5`. Regular Convex Hexagon: This type combines the characteristics of both regular and convex hexagons. It has equal side lengths, and equal angles, and all its interior angles are less than `180°`. It exhibits the symmetry and uniformity of a regular hexagon while maintaining convexity.
`6`. Regular Concave Hexagon: A regular concave hexagon possesses the properties of both regular and concave hexagons. It has equal side lengths, equal angles, and at least one interior angle greater than `180°`. Despite its concavity, it maintains the symmetry and uniformity of a regular hexagon.
Understanding Hexagon Sides
A hexagon consists of six straight sides that come together to form a closed shape. Whether it's a regular or irregular hexagon, these sides create the boundaries of the shape. In a regular hexagon, all six sides are equal in length, while in an irregular hexagon, at least two sides have different measures.
To find the perimeter of a hexagon, simply add together the lengths of all its sides.
For a regular hexagon, if you know the perimeter, you can easily calculate the length of each side by dividing the perimeter by six. For instance, if the perimeter of a regular hexagon is `90` units, then each side measures `90 ÷ 6 = 15` units.
Understanding Angles in a Regular Hexagon
In a regular hexagon, each angle inside the shape measures `120°`. This can be found by dividing the total sum of interior angles, which is `720°`, by the number of angles, which is `6`.
Moreover, the exterior angles of a regular hexagon measure `60°` each. This is calculated by dividing the total sum of exterior angles, which is `360°`, by the number of angles, which is also `6`.
Understanding Angles in an Irregular Hexagon
In an irregular hexagon, the angles and side lengths can vary. Each interior angle measure depends on the specific shape. To find the sum of interior angles, add up all angles within the hexagon.
Similarly, each exterior angle measure can differ. To find an exterior angle, subtract the adjacent interior angle from `180°`. Understanding angles in irregular hexagons involves examining the shape's characteristics, making it a bit complex compared to regular hexagons.
Perimeter of a Hexagon
The perimeter of a hexagon is the total length of its six sides. Whether it's a regular or irregular hexagon, calculating the perimeter involves adding together the lengths of all sides.
In a regular hexagon where all sides are equal, you can find the perimeter by multiplying the length of one side by six. For an irregular hexagon, where side lengths may vary, you simply sum up the lengths of all six sides.
Properties of a Hexagon
A hexagon is a polygon with six sides and six angles. Understanding its properties is crucial in geometry and various real-world applications.
`1`. Number of Sides and Angles: A hexagon has six sides and six angles.
`2`. Sum of Interior Angles: The sum of interior angles in a hexagon is always `720°`. This remains true for both regular and irregular hexagons.
`3`. Sum of Exterior Angles: The sum of exterior angles in any polygon, including a hexagon, is always `360°`.
`4`. Types of Hexagons: Hexagons can be regular, with all sides and angles equal, or irregular, with varying side lengths and angles.
`5`. Symmetry: A regular hexagon has six lines of symmetry, each dividing it into equal halves. Irregular hexagons may have fewer lines of symmetry or none at all.
`6`. Diagonals: A hexagon has nine diagonals three long diagonals and six short diagonals.
`7`. Area: The area of a hexagon can be calculated using various methods, such as splitting it into triangles or using specific formulas especially based on side lengths. One common formula for the area of a regular hexagon is:
`\text{Area} = ((3\sqrt{3}) / 2) * s^2`, where `s` is the length of one side.
Diagonals of a Hexagon
In a hexagon, diagonals are line segments connecting non-adjacent vertices (corners) of the polygon. Understanding the diagonals of a hexagon is essential in geometry and various mathematical applications.
`1`. Number of Diagonals: A hexagon has a total of nine diagonals. These diagonals can be categorized into three long diagonals, which connect opposite vertices, and six short diagonals, which connect non-adjacent vertices.
`2`. Long Diagonals: The long diagonals of a hexagon are the line segments that connect opposite vertices of the shape. There are three long diagonals in a hexagon, passing through the center of the hexagon. The length of a long diagonal of a regular polygon in `2a`, where `a` is the side length of the regular polygon.
`3`. Short Diagonals: The short diagonals of a hexagon are the line segments that connect non-adjacent vertices of the shape, without passing through the center of the polygon. There are six short diagonals in a hexagon, forming smaller triangles within the polygon. The length of a short diagonal of a regular polygon is `sqrt3a`, where `a` is the side length of the regular polygon.
Solved Examples
Example `1`. If the perimeter of a regular hexagon is `30` `\text{cm}`, what is the length of each side?
Solution:
The perimeter of the regular hexagon `= 30` `\text{cm}`
Since a regular hexagon has six equal sides, we can find the length of each side by dividing the perimeter by `6`.
`"Length of each side" = (\text{Perimeter}) / 6 = "30 cm"/ 6 = "5 cm"`
Therefore, the length of each side of the regular hexagon is `5` `\text{cm}`.
Example `2`. The interior angles of a regular hexagon are each `120` degrees. What is the sum of all interior angles of the hexagon?
Solution:
Each interior angle of the regular hexagon `= 120` degrees
To find the sum of all interior angles, we multiply the measure of one interior angle by the number of angles in the hexagon.
`"Sum of interior angles" = "120 degrees" * 6 = "720 degrees"`
Therefore, the sum of all interior angles of the regular hexagon is `720` degrees.
Example `3`. If the length of one diagonal of a hexagon is `8` cm, what is the total length of all diagonals in the hexagon?
Solution:
Given: `"Length of one diagonal"` `=` `"8 cm"`
A hexagon has a total of nine diagonals. Three of these are long diagonals, and six are short diagonals.
`"Length of a long diagonal" = 2 * 8 = "16 cm"`
`"Length of a short diagonal" = \sqrt{3} * 8 = 8\sqrt{3}" cm"`
\(
\begin{align*}
\text{Total length of all diagonals} &= 3 \times \text{length of long diagonal} + 6 \times \text{length of short diagonal} \\
&= 3 \times 8 \text{ cm} + 6 \times 8\sqrt{3} \text{ cm} \\
&= 24 \text{ cm} + 48\sqrt{3} \text{ cm} \\
&= 72 \text{ cm} + 83.14 \text{ cm} \\
&= 155.14 \text{ cm}
\end{align*}
\)
Therefore, the total length of all diagonals in the hexagon is `"155.14 cm"`.
Example `4`. The length of the long diagonal of a regular hexagon is `"12 cm"`. What is the length of the short diagonal?
Solution:
Given: `"Length of long diagonal"` `=` `"12 cm"`
`"So, the side length of the regular hexagon" = 12/2 = "6 cm"`
Hence, `"the measure of the short diagonal" = 3 * 6 = "10.39 cm"`
Practice Problems
Q`1`. The sum of exterior angles in a hexagon is ______ degrees.
- `540`
- `600`
- `720`
- `360`
Answers: d
Q`2`. In a regular hexagon, all interior angles are equal.
- True
- False
Answers: a
Q`3`. If the perimeter of a regular hexagon is `42` `\text{cm}`, what is the length of each side?
- `6` `\text{cm}`
- `7` `\text{cm}`
- `8` `\text{cm}`
- `9` `\text{cm}`
Answers: b
Q`4`.The length of each side of a regular hexagon is `5` `\text{cm}`. What is the perimeter of the hexagon?
- `20` `\text{cm}`
- `25` `\text{cm}`
- `30` `\text{cm}`
- `35` `\text{cm}`
Answers: c
Frequently Asked Questions
Q`1`. What is a hexagon?
Answer: A hexagon is a polygon with six sides and six angles. It is a two-dimensional shape commonly found in geometry and nature.
Q`2`. How many angles does a hexagon have?
Answer: A hexagon has six angles. Each interior angle measures `120` degrees in a regular hexagon.
Q`3`. What is the sum of interior angles in a hexagon?
Answer: The sum of interior angles in any hexagon is always `720` degrees. This remains true regardless of whether the hexagon is regular or irregular.
Q`4`. What are the properties of a regular hexagon?
Answer: A regular hexagon has six equal sides and six equal angles. It possesses symmetry, with six lines of symmetry dividing it into congruent parts.
Q`5`. How do you find the perimeter of a hexagon?
Answer: To find the perimeter of a hexagon, you add the lengths of all six sides. In a regular hexagon, where all sides are equal, you can multiply the length of one side by six.
Q`6`. What is the difference between a regular and irregular hexagon?
Answer: A regular hexagon has all sides and angles equal, while an irregular hexagon has sides and/or angles of different lengths or measures.
Q`7`. How can I calculate the area of a hexagon?
Answer: The area of a hexagon can be calculated using various methods, such as dividing it into triangles or using specific formulas based on side lengths and apothem. One common formula for the area of a regular hexagon is:
`\text{Area} = ((3\sqrt{3}) / 2) * s^2`, where `s` is the length of one side.