Calculate the volume and the total surface area of a cone of base diameter 16cm and slant height 17cm in terms of pie
The radius of the cone, r = 8 cm.
Using Pythagoras theorem, we can find the height of the cone:
height^2 + radius^2 = slant height^2
height^2 + 8^2 = 17^2
height^2 = 17^2 - 8^2
height^2 = 225
height = 15 cm
Volume of cone = (1/3)πr^2h = (1/3)π(8^2)(15) = 320π/3 cubic cm
Total surface area of cone = πr^2 + πrℓ = π(8^2) + π(8)(17) = 288π/3 + 136π/3 = 424π/3 square cm.
To calculate the volume of a cone, you can use the formula:
Volume = (1/3) * π * r^2 * h
Where:
- π is a mathematical constant approximately equal to 3.14159
- r is the radius of the cone (half of the base diameter)
- h is the height of the cone (slant height minus the radius)
Given:
- Base diameter = 16cm
- Slant height = 17 cm
- π (pi) is approximately 3.14159
Calculating the radius:
Radius = (base diameter) / 2 = 16cm / 2 = 8cm
Calculating the height (h):
h = slant height - r = 17cm - 8cm = 9cm
Plugging these values into the formula for the volume of a cone:
Volume = (1/3) * π * (8cm)^2 * 9cm
Volume = (1/3) * 3.14159 * 64cm^2 * 9cm
Volume ≈ 602.88cm^3 (rounded to two decimal places)
To calculate the total surface area of a cone, you can use the formula:
Total Surface Area = π * r * (r + slant height)
Plugging the given values into the formula:
Total Surface Area = 3.14159 * 8cm * (8cm + 17cm)
Total Surface Area ≈ 528.146cm^2 (rounded to three decimal places)
Therefore, the volume of the cone is approximately 602.88 cubic cm and the total surface area is approximately 528.146 square cm.