Find side length, cirumference and area of octagon

• MHB
• Daugava
In summary, there are various formulas and methods for finding the side length, circumference, and area of an octagon. These include using the radius of the circumscribed circle, dividing the circumference or area by the number of sides, and using specific formulas for each measurement. Additionally, there are online tools and software programs that can quickly calculate these values for you.
Daugava
Square-shaped piece of paper is intended to make a regular octagon through the cutting of the vertices of a square.
The length of the piece of paper is 29 cm.
How long triangle cathetus have to be cut off from the vertices of the square?
Calculate the octagonal side length, circumference, and area.
Sorry for my english, if it's not understandable I'll try to explain it better.. but how do I do this?
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Daugava said:
Square-shaped piece of paper is intended to make a regular octagon through the cutting of the vertices of a square.
The length of the piece of paper is 29 cm.
How long triangle cathetus have to be cut off from the vertices of the square?
Calculate the octagonal side length, circumference, and area.
Sorry for my english, if it's not understandable I'll try to explain it better.. but how do I do this?
Hi Daugava, and welcome to MHB!

You want to cut four triangles from the square of paper. Suppose that the shorter sides of these triangles have length $x$ cm. Then the hypotenuse of the triangle will be $\sqrt2x$ cm. After cutting off the triangles, the sides of the square will have been shortened by $2x$ cm. You want all the sides of the resulting octagon to have the same length. That implies that $\sqrt2x = 29 - 2x$. Solve that equation to find the octagonal side length. It should then be easy to find the circumference of the octagon. For the area of the octagon, subtract the area of the four triangles from the area of the square.

So the side length is 12 cm
a = 12 cm
Area is 2*(1+sqrt2)a^2 = 695,3 cm^2
Circumference is 8*a = 96 cm
Is this right?
But how do I find out how long the triangle cathetus/sides are that were cut off from the vertices of the square?
Is the cut off 8,5 cm?

Last edited:
Daugava said:
So the side length is 12 cm
a = 12 cm
Area is 2*(1+sqrt2)a^2 = 695,3 cm^2
Circumference is 8*a = 96 cm
Is this right?
But how do I find out how long the triangle cathetus/sides are that were cut off from the vertices of the square?
Is the cut off 8,5 cm?
Thank you for teaching me a new word in my own language! I did not know that a cathetus is one of the perpendicular sides of a right-angled triangle.

The side length is not precisely 12. In fact, it is $29(\sqrt2-1)$, which is approximately 12.012. The perimeter is approximately 96.0975.

The value for the cathetus comes from my previous comment above. If the cathetus is $x$ cm, then $\sqrt2x = 29 - 2x$. Solve that equation to get $x = \dfrac{29}{2+\sqrt2}.$

1. How do you find the side length of an octagon?

The formula for finding the side length of an octagon is s = r * √2, where s is the side length and r is the radius of the circumscribed circle. You can also divide the circumference of the octagon by 8 to get the side length.

2. How do you calculate the circumference of an octagon?

The formula for finding the circumference of an octagon is C = 8s, where C is the circumference and s is the side length. Alternatively, you can also use the formula C = 2πr, where r is the radius of the circumscribed circle.

3. How do you determine the area of an octagon?

The formula for finding the area of an octagon is A = 2(1+√2) * s^2, where A is the area and s is the side length. You can also divide the octagon into smaller shapes, such as triangles or rectangles, and use the appropriate formulas to find the area.

4. Can you find the side length, circumference, and area of an octagon if you only know the radius?

Yes, you can use the formula s = r * √2 to find the side length, C = 2πr to find the circumference, and A = 2(1+√2) * r^2 to find the area if you know the radius of the circumscribed circle.

5. Is there a faster way to find the side length, circumference, and area of an octagon?

Yes, there are several online calculators and software programs that can quickly and accurately calculate the side length, circumference, and area of an octagon. You can also create your own spreadsheet or program using the formulas mentioned above for faster calculations.

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