find the are of a regular octagon whose sides are 12cm long.
cut the octogon into 8 congruent triangles, with vertices at the centre of the octogon.
Consider one of those triangles.
It will have a base of 12 cm, and the angle across from the 12 will be 45º, with the other two angles 67.5º each.
Let its height be h.
tan 67.5 = h/6
h = 14.485
so the area of that triangle is 1/2(12)(14.485
= 86.912
so the octogon is 8(86.912) = 695.29 cm^2
To find the area of a regular octagon, you can follow these steps:
Step 1: Divide the octagon into smaller isosceles triangles.
Step 2: Find the area of one isosceles triangle.
Step 3: Multiply the area of one triangle by 8 to get the total area of the octagon.
Let's go through these steps in more detail.
Step 1: Divide the octagon into smaller isosceles triangles.
A regular octagon can be divided into eight congruent isosceles triangles by drawing lines from the center of the octagon to each vertex. This creates eight identical triangles with one side as the base and the other two sides as the equal legs.
Step 2: Find the area of one isosceles triangle.
Since the sides of the octagon are 12 cm long, the base of each isosceles triangle is 12 cm. To find the height of the triangle, we can use the formula for the height of an isosceles triangle: h = √(leg^2 - (base/2)^2). Since each leg of the isosceles triangle is equal to half of a side of the octagon, we can substitute the values into the formula: h = √(12^2 - (12/2)^2).
h = √(144 - 36) = √108 ≈ 10.39 cm (rounded to two decimal places)
Now that we have the base and height of the triangle, we can use the formula for the area of a triangle: A = (base * height) / 2.
A = (12 * 10.39) / 2 = 124.68 / 2 = 62.34 cm² (rounded to two decimal places)
Step 3: Multiply the area of one triangle by 8.
Since there are eight identical triangles in the octagon, we can multiply the area of one triangle by 8 to get the total area of the octagon.
Total Area = 62.34 cm² * 8 = 498.72 cm² (rounded to two decimal places)
Therefore, the area of the regular octagon with sides 12 cm long is approximately 498.72 cm².
To find the area of a regular octagon, you can use the formula:
Area = 2 * (1 + √2) * s^2
Where "s" represents the length of each side.
In this case, the length of each side is given as 12 cm. Therefore, you can substitute this value into the formula and calculate the area as follows:
Area = 2 * (1 + √2) * (12 cm)^2
= 2 * (1 + √2) * 144 cm^2
≈ 387.31 cm^2
So, the area of the regular octagon with sides measuring 12 cm is approximately 387.31 square centimeters.