Find the surface area of a right octagonal pyramid with height 2.5 yards, and its base has apothem length 1.5 yards.
Surface Area=Base Area+Lateral Area
SA=BA+LA
The side of the octagon
a=2*apothem*tan(45/2)
BA=0.5*8a*apothem=8*apothem^2*tan(45/2)
LA=4a*(slant height)
slant height=sqrt(height^2+apothem^2)
tan(45/2)=0.4142
To find the surface area of a right octagonal pyramid, we need to calculate the areas of its individual components and then add them together.
First, let's find the area of the octagonal base. The formula to calculate the area of a regular octagon is:
Area = 2 * (1 + √2) * a²,
where 'a' is the apothem length of the octagon. In this case, the apothem length is given as 1.5 yards. Plugging this value into the formula, we have:
Area of base = 2 * (1 + √2) * (1.5²).
Next, let's calculate the lateral area of the pyramid. The formula to find the lateral area of a pyramid is given by:
Lateral area = 1/2 * perimeter of base * slant height,
where the slant height is the distance from the apex (tip) of the pyramid to the midpoint of any of the triangular faces. The perimeter of the octagonal base is given by:
Perimeter of base = 8 * a,
where 'a' is the apothem length. Plugging in the given apothem length value, we have:
Perimeter of base = 8 * 1.5.
To find the slant height, we can use the Pythagorean theorem. In this case, the height of the pyramid is given as 2.5 yards, and the base of the triangle formed by the slant height, height, and a side of the octagon is an isosceles right triangle. Therefore, the slant height can be found as:
Slant height = √(height² + (a/√2)²).
Substituting the given values into the formula, we have:
Slant height = √(2.5² + (1.5/√2)²).
Finally, we can calculate the lateral area by plugging in the values we found:
Lateral area = 1/2 * (8 * 1.5) * √(2.5² + (1.5/√2)²).
Once we have the area of the base and the lateral area, we can find the total surface area by summing them:
Surface area = Area of base + Lateral area.
By substituting the values we found into this equation, we can calculate the surface area of the right octagonal pyramid.