Find the surface area of a right octagonal pyramid with height 2.5 yards, and its base has apothem length 1.5 yards.
Correction!
No life
The surface area is equal to the area of the base plus the area of the 8
isosceles triangles on the sides.The base consist of 8 isosceles triangles
also.
A = Ab + A(sides).
Ac = 360/N = 360/8 = 45Deg = central angle.
Base angles = (180 - 45) / 2 = 67.5deg
each.
tan67.5 = h/(b/2) = 1.5/(b/2),
b/2 = 1.5 / tan67.5 = 0.62,
b = 2 * 0.62 = 1.24yds = base of each
triangle.
Ab = (b*h/2)N = (1.24*1.5/2)8 = 7.44sq
yds.
Area of Sides:
Altitude = 2.5*sin67.5 = 2.31YDS.
Area of sides = (b*h/2)N =
(1.24*2.31/2)8 = 11.5 sq. yds.
Surface Area = 7.44 + 11.5 = 18.9 sq yds.
Well, if you start with a right octagonal pyramid, it's like a pyramid with an octagon for its base. So let's get circus-y and break it down, shall we?
First, let's calculate the area of the base. Since we have an octagon, the formula for the area is given by A = 2 * (1 + √2) * a^2, where "a" is the apothem length (1.5 yards in this case). But since we want the surface area, not just the area of the base, we have to add the area of the eight triangular faces.
Each triangular face of the pyramid is an isosceles triangle, with its two equal sides being half the length of one side of the octagon base (that's √2 * a). The height of the triangle is equal to the height of the pyramid (2.5 yards).
Now, let's calculate the area of one of these triangular faces: (1/2) * (√2 * a) * height.
Adding the area of all eight triangular faces to the area of the base, we get the total surface area of the pyramid: 8 * (1/2) * (√2 * a) * height + A.
Plug in the values, do the math, and you'll have the surface area of that octagonal pyramid. I guarantee it's going to be a real hit at the circus!
To find the surface area of a right octagonal pyramid, we need to calculate the sum of the areas of all its faces.
First, let's identify the different faces of the pyramid. A right octagonal pyramid has one base and eight triangular faces.
The base of the pyramid is an octagon, which has eight equal sides. Each of these sides can be divided into two parts: the radius (apothem) and the slant height. In this case, the apothem length is given as 1.5 yards.
To calculate the slant height, we can use the Pythagorean theorem. We know that an octagon is made up of eight isosceles triangles with a right angle at the center. The hypotenuse of each triangle is the slant height. The other two sides are the radius (apothem) and half the length of one side of the octagon.
Since the octagon's sides are equal, we can find the length of one side by dividing the total length of all eight sides by eight. However, the given information doesn't provide the length of the sides, so we can't directly calculate the slant height.
Without the length of the sides, we cannot continue to calculate the slant height or the surface area.
Please provide the length of one side of the octagon to proceed with the calculation.
CORRECTION!
Area of the sides:
The height(altitude) is GIVEN: 2.5 yds.
S = h/sin67.5 = 2.5 / sin67.5 - 2.71
yds = length of each side.
As = (b*h/2)N = (1.24*2.5/2)8 = 12.4sq yds.= Area of sides.
Surface Area = Ab + As = 7.44 + 12.4
= 19.84sq yds.