0
How do you find the apothem of an octagon with only the side length?
Wiki User
∙ 14y ago
I wish I could draw or insert an octagon on this page.
You draw an octagon, or do a Google search for octagon and copy paste the image of the octagon on MS word. Set it so one side is horizontal. There should be another horizontal side on the top of the octogon.
Draw an x by drawing line from the left end point of the top side to the right end point of the bottom side. Draw another line from the right end point of the top side to the left end point of the bottom side. The 2 lines intersect at the center of the octagon. If you continue drawing these lines you will have 8 triangles, each with a vertex angle equal to 1/8 of 360o. So, the vertex angle of each of the 8 isosceles triangles is 45o
Look at the triangle whose base is the bottom side of the octagon The vertex angle at the top of this isosceles triangle is 45o. The sum of the interior angles of a triangle equals 180 o. The two base angles are equal. Let x equal the degrees of one of the base angles of the isosceles triangle.
2x + 45o = 180o
2x = 135o
x = 67.5o
Draw a line straight down from the center point of the octagon to the base of the triangle (bottom of octagon). This vertical line is the apothem. The apothem (vertical line) cuts the isosceles triangle into 2 right triangles. Now we can use trigonometry. Looking at the right triangle on the left side of the apothem, you know the left, bottom angle equals 67.5o and the apothem is the side opposite that angle.
Tangent = Opposite ÷ Adjacent
Adjacent is ½ of the length of the side of the octagon.
Opposite is the apothem
Example:
Side length = 10cm
½ side length = 5 cm
Angle = 67.5o
Tangent 67.5o = Opposite ÷ 5cm
Opposite = Tangent 67.5o × 5cm
Opposite = 12.07 cm
Or
Since the apothem is perpendicular to the base of the isosceles triangle and bisects the base, it also bisects the vertex angle in to two angles of 22.5o. Therefore, in the right triangle with base the one half of the side s of the octagon and height the apothem a we have:
a = (s/2)cot 22.5o
In the above example,
a = 5 cm (cot 22.5o) ≈ 12 cm
Wiki User
∙ 14y ago
Add your answer:
Earn +20 pts
How do you find height of isosceles triangle if you only know the length of equal sides and the base?
i can
How do you find the area of a square given only the length of its diagonal?
Given the length of the diagonal of the square ... call it 'D units'. The area of the square is (1/2 D2) (same units)2.
How do you find the area of a rectangle with only the width?
Measure...is the length twice the width or 3 times. If so, then just multiply that amount, then figure like normal.
How do you find the length of a side of a rectangle when you only have the area?
It can't be done. You must also know at least any one of the following: Perimeter Relation between length and breath Relation between Area and length Relation between Area and breath Relation between perimeter and Area Breath and so on...........
Draw an angle with 370 degrees?
Angles only go up to 360 degrees unless it's 1 and 1/36 of a revolution
How do you find the apothem of an octagon with the diameter?
You can only do this for a regular octagon. It is much easier to understand the method if you do a rough sketch and follow the explanation using that. Unfortunately, this browser does not support any kind of drawing! Suppose the diameter of the octagon is D. Therefore the diameter of the circumscribing circle is also D. Form the centre of this circle, draw lines to two adjacent vertices of the octagon. The lengths of these lines is D/2 because these are radii of the circle. These lines and the side of the octagon form an isosceles triangle, and the apothem is the height of this triangle. Now consider half this triangle: the right angled triangle formed by the apothem, half the side of the octagon and the radius. The angles at the apex of the octagon is 360/8 = 45 degrees. So the angle at the apex of the right angled triangle is half that = 22.5 degrees. Then cos(22.5 deg) = Apothem/Radius So that Apothem = Radius*cos(22.5 deg) = D/2*0.9239 (approx).
How do you find the area of a octagon with only the apothem?
Octagon is composed of 8 triangles, which can be combined in pairs to make 8 rectangles of which one side is the apothem. The area is 1.657 x the square of the apothem, simplified from 4 times the square of the apothem divided by tan67.5o ie 2.4142. 4 divided by 2.4142 is 1.657. Consider triangle OAB where OB is the apothem and AB is half of a side: The angle OAB = 67.5 degrees, half of the interior angle of a regular hexagon. Tan 67.5 = OB/AB so AB = OB/tan 67.5 and the area of each notional rectangle is OB squared divided by tan 67.5. There are four such rectangles hence 4OB2/tan 67.5 is the area of the octagon.
How do you find the area of a hexagon without a side length?
To find the area of a Regular hexagon with side length (x) you need:1. The "radius" of the hexagon. (Just the length from the center to the outside edge.)2. The apothem. (which is only just half of the height of the base.)**If you don't have one or both of these you can't do it.**Steps:1. Make a triangle of the apothem (used as a) and the radius. (r)2. Use the Pythagorean Theorem to find 1 half of the side length.3. Multiply the actual side length by 6.4. Multiply that by a.5. The area is your answer.
Is an octagon a polyhedron?
Adding to what Kittenono said, an octagon is a two dimensional shape which only has length and width. A polyhedron is 3 dimensional which has length, width and height.
How do you find the apothem of a hexagon with only the side length?
The apothem is the radial distance from the middle of the side to the centre of the hexagon. A hexagon is six congruent equilateral triangles joined by adjacent sides. Equilateral triangles can be divided into two equal right angled triangles. The upright of the right angled triangle is effectively the apothem of the original hexagon. Pythagoras now kicks in. The apothem (vertical) is A and half the side length (base) is B, the third (longest) side is C and is the same as the original side length. Pythagoras states A2 + B2 = C2. So by transposition, A = root (C2 - B2). As B = 1/2 C, the apothem A is given by: A = root(C2 - (C/2)2) = root(C2 - C2/4) = root(3C2/4) = C x root(3) / 2 So the apothem of a hexagon is 1/2 x root(3) x the side length.
How do you calculate the area of the polygon?
If it is a regular polygon--meaning that all the sides are congruent and all the angles are congruent, then the formula for area of the polygon is A=1/2 ap Here a represents the apothem, which is the distance from the center of the polygon to the midpoint of one side. p represents the perimeter of the polygon found by multiplying one side length by the number of sides. If you only know one variable such as side length, you can find the perimeter and you can find the apothem using trigonomety.
Is the perimeter of an octagon 64?
The perimeter of an octagon is 64 only if it is a regular octagon and the length of one side is 8. An irregular octagon may or may not have a perimeter of 64 (whatever units of measure you choose to use) depending on what the lengths of the individual sides are.
How do you find the area of an octagon when you only know the apothegm?
ask. mrs.cwiklinski
Is a octagon equilateral?
Only if all the SIDES are EQUAL in length. Equi -> Equal Lateral -> Sides To actually answer the question: An octagon has all sides the same length and its internal angles are all the same size. All regular polygons except for a triangles are equilateral.
What is the Area of an octagon with sides 7.5 in. and diameter of 8.25 in.?
What is the "diameter" of an octagon? For a circle it is the length of the chord between two points on the circumference which passes through the centre of the circle. This can also be described as the length of a straight line between two points on the circumference of the circle which also passes through the point which is the centre of the circle. For an octagon you cannot choose a point such that the length of the straight line between two points on opposite sides of the octagon which passes through that point is a constant. There is a circumcircle which is the circle which passes through all vertices of the octagon. When all sides of the octagon are equal in length this can only occur if the octagon is regular (ie every interior angle of the octagon is the same [at 135°]). In this case the diameter of this circle for an octagon with side length 7.5 in is approx 19.6 in There is also the inscribed circle which is the largest circle that can be drawn inside the octagon. If it touches every side, the octagon must be a regular octagon and this circle has a diameter of approx 18.1 in. If you are referring to a circle with a diameter of 8.25 in drawn between two opposite sides (which are parallel) then this does not define a unique octagon as the octagon will be rather squashed and the three sides at each end of the opposite sides which are 8.25 in apart can be placed in up to two different configurations which lead to 4 different possible areas for each skew position of the sides 8.25 in apart. --------------------------------------------------------------------------------------------- If you are asking for the area of the regular octagon with side length 7.5 in then: Consider the circumcircle of the octagon and its centre O. Join every vertex to the centre O. The area of the octagon is the sum of the areas of these triangles. As two sides of each the triangle are radii of the circle, they are isosceles triangles. As the third side of each triangle is the same, these isosceles triangles are congruent. The area of the regular octagon is thus 8 times the area of one of these triangles. As there are 8 triangles all around the centre O, the angle at the vertex at the point O is: Angle at O = 360° ÷ 8 = 45° Drop a perpendicular from this angle to the side of the octagon to get the height of the triangle As the triangle is an isosceles triangle, this perpendicular line bisects the angle at O, and the base of the triangle (side of the octagon). You now has a right angle triangle with its hypotenuse the radius of the circumcircle, one side half the length of the side of the octagon and its third side the height of the isosceles triangle. Using the tangent ratio, the height of the isosceles triangle can be found: tangent = opposite/adjacent → tan 22½° = half_octagon_side_length / height_of_triangle → height_of_triangle = (7.5 in ÷ 2) / tan 22½° ≈ 9.05 in → area triangle ≈ ½ × 7.5 in × 9.05 in ≈ 33.95 in² → area octagon = 8 × area triangles ≈ 8 × 33.95 in² ≈ 271.6 in² ------------------------------------------------------------------------------------------ This can be generalised into the area of ANY regular polygon: area_regular_polygon = side² × ¼ × number_of_sides ÷ tan(180° ÷ number_of_sides) For an octagon with side length 7.5 in, this gives: area = (7.5 in)² × ¼ × 8 ÷ tan(180° ÷ 8) ≈ 271.6 in² as before. The height of the isosceles triangles is called the apothem of the [regular] polygon; it is the radius of the inscribed circle. This means the formula for the area of the regular polygon becomes: area_regular_polygon = number_of_sides × ½ × side × apothem
What is the area of a regular hexagon with a side length of 4.5 m and an apothem length of 3.9 m?
Given the side length of a regular polygon you don't need the apothem and conversely. Using the side length, you get an area of 52.61 sq metres. Using the apothem, you get 52.69 sq metres. The difference between the two is because, if the side were exactly 4.5 m, then the apothem would be side*sqrt(3)/2. This is 3.897 m (to 3 dp) rather than 3.9. The difference leads to the answer for the area being 52.6 or 52.7 sq metres. However, the lengths are given to only 2 significant digits and to 2 sig digs, the area is 52 sq metres. This is a good example for people who insist on giving answers to spurious levels of accuracy.
What is unique about the octagon?
Only the octagon has 8 sides.
