Cracking the Octagon Area Code: Geometry and Practical Applications
Comprehensive Definition, Description, Examples & RulesÂ
Introduction to Octagon Area
Polygons with 8 sides are known as octagons. Octagons are used in a variety of real-life applications like geometry and design. In geometry. It comes in handy in calculating the space occupied by shapes with 8 sides and in design and artwork, it helps create octagonal shapes like windows, lenses, etc.Â
Understanding the Octagon
An octagon is a polygon with a total number of sides of 8. Here are some unique properties of the 8-sided figure:
- An octagon always has 8 sides in total.
- An octagon always has 8 angles.
- All the interior angles in an octagon sum up to 1080 degrees.
- All the exterior angles in an octagon sum up to 360 degrees.
- Every interior angle in a regular octagon measures 135 degrees.
- Every exterior angle in a regular octagon measures 45 degrees.
- There are no more than 20 diagonals in an octagon.Â
The Octagon Area Formula
The Area of a regular octagon follows the given formula for an accurate computation:Â
A= 2 * (1+2) * s^2
A: Octagon’s area
s: Measure of one of the sides of a regular octagon
Finding the Area of an Octagon
Here is a step-by-step strategy of how you can discern the area of a regular octagon, with the help of the area of an octagon formula given above:Â
- The Area of a regular octagon follows the given formula for an accurate computation: A= 2 * (1+2) * s^2 (A: Octagon’s area, and s: Measure of one of the sides of a regular octagon)
- Use the values available to you and place them in the equation of the formula
- Carry out calculations and mathematical functions to get the final answer
Example: What will be the area of a regular octagon which has a side measuring up to 8 cm?
Solution: A= 2 * (1+2) * s^2Â
A= 2 * 2.41 * 8^2
A= 4.82 * 64
A= 308.48 cm^2
Special Cases: Finding Area Without Height
Suppose you do not know the height of a trapezoid or an octagon, you can still know the area using other methods like
- If you know the length of the base of the trapezoid or octagon and that of the non-parallel side, you can use the formula ½(b1+b2)s to calculate the area of the octagon where b1 and b2 are the lengths of the two bases and s is the length of the non-parallel side.
- If you know the length of the base of the trapezoid or octagon and that of the diagonals, you can use the formula ½(b1+b2)d to calculate the area of the octagon where b1 and b2 are the lengths of the two bases and d is the average of the two diagonals of the octagon.
Real-World Applications
The knowledge of octagons is used in a variety of real-world applications, for example:
- In architecture, octagons are used to create various items of everyday use like octagonal windows, lenses, etc.Â
- In construction, octagons are used to design roads and highways and do concrete work.Â
- In geometry, too, octagons have various uses as we can know how much space an 8-sided shape covers or its boundary length if we know octagons.Â
Infographics: Visual Representation
Here is a visual representation of a trapezoid or octagon to help you better understand the octagon area formula:Â
Trapezoid Area vs. Other Shapes
Different formulas are used to compute the areas of different geometrical shapes. Here is a representation of the different kinds of shapes found commonly and their respective formulas for computing their area:
Trapezoid
The area of a trapezoid or an octagon can be computed with the formula:
A= 2 * (1+2) * s^2 (A: Area of octagon, s: length of the side of the octagon)
Rectangles
The area of rectangles is computed with the help of the formula:Â
A= l * b,Â
‘A’: area of the rectangle.
‘l’: length of one of the sides of the rectangle.
‘b’ refers to the breadth of one of the sides of the rectangle.
Parallelograms
A parallelogram’s area can be determined using the formula:Â
A= b * h
 ‘A’: the area of the rectangle.
‘h’ is used to denote the height of the parallelogram.
‘b’ refers to the measure of the base of the parallelogram.
Common Mistakes and Pitfalls
Here is a list of some common errors when computing the area of octagons and what you should do to avoid them:
- Often, people read the measurements given incorrectly. Make sure to double-check the measures available to you before starting your calculations.
- People also make the mistake of using miscellaneous units throughout the formula, resulting in an inaccurate result. Ensure to maintain a unity of units in the formula for the most accurate and precise solutions.Â
- People often confuse the area formula for octagons with other geometrical shapes. Memorize the particular area formulas for each geometrical shape to carry out your calculations quickly and fool-proof.Â
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Key Takeaways
- An octagon is a polygon with a total number of sides of 8.
- The formula for computing the area of a regular octagon is given as: A= 2 * (1+2) * s^2 (A: Area of octagon, s: length of the side of the octagon)
- Octagons are used in a variety of fields like architecture, geometry, construction, and more.
Quiz
Question comes here
Frequently Asked Questions
For times when you don’t know the measure of a side of a trapezoid or an octagon, you can still calculate the octagon area by taking the measure of the total distance from the central point of the octagon to the central point of any side (also called the apothem), and then multiply it with a half and the perimeter of the octagon (sum of the length of all sides).Â
A= ½ * apothem * perimeter
In architecture and design, octagons create windows, roofs, lenses, etc.Â
The area of an octagon is represented as 2 * (1+√2) * s^2, which multiplies the square of the length of the side with (1+√2) and then with 2. The area of a triangle is represented as ½*b*h, which takes the base and the height of the triangle into consideration, and that of a hexagon is represented as 3√3/2*s^2, which multiplies the square of the length of the side with the half of 3√3.
Yes, octagons have various unique properties that influence their area calculations, like:
- There are always eight sides to an octagon.
- There are always eight angles in an octagon.
- An octagon’s internal angles cannot be greater than 1080 degrees in total.
- In an octagon, the sum of all external angles cannot exceed 360 degrees.
- In a standard octagon, each inner angle is 135 degrees.
- In a standard octagon, each outer angle is 45 degrees.
- An octagon can have no more than 20 diagonals.Â
Using the incorrect formula and using inconsistent units are two typical errors to prevent when computing the area of an octagon. To prevent misunderstanding, remember the multiple area calculations for different geometric forms and utilise the same units of measurement throughout the formula.
