A cone and a square based pyramid have slant heights of 6 inches . The diameter for the cone and the base edge of the pyramid are both 8 inches.
How would I find which space figure has the greater surface area
rifyuxj
To determine which space figure has the greater surface area between the cone and the square-based pyramid, we can use the formula for surface area for each figure and compare the results.
Let's start with the cone:
The formula for the surface area of a cone is given by:
SA_cone = π * r * (r + l)
where r is the radius of the base and l is the slant height.
In this case, the diameter of the cone is given as 8 inches, which means the radius (r) is half of that:
r = 8 / 2 = 4 inches
The slant height (l) is given as 6 inches.
Now we can calculate the surface area of the cone:
SA_cone = π * 4 * (4 + 6) = 10π + 4π = 14π square inches
Next, let's move on to the square-based pyramid:
The formula for the surface area of a square-based pyramid is given by:
SA_pyramid = l * (s + 2a)
where l is the slant height, s is the length of one side of the base, and a is the apothem (the perpendicular distance from the center of the base to one of the triangular faces).
In this case, the slant height (l) is given as 6 inches.
The base edge of the pyramid is also given as 8 inches, which means the length of one side of the base (s) is equal to that:
s = 8 inches
In a square, the apothem (a) is half the length of the side, so:
a = s / 2 = 8 / 2 = 4 inches
Now we can calculate the surface area of the square-based pyramid:
SA_pyramid = 6 * (8 + 2 * 4) = 6 * (8 + 8) = 6 * 16 = 96 square inches
Comparing the surface areas of the cone and the square-based pyramid, we have:
SA_cone = 14π square inches
SA_pyramid = 96 square inches
Since we don't have an exact value for π, we can approximate it as 3.14. By comparing the numerical values, we can see that 14π is greater than 96. Therefore, the cone has the greater surface area.