Calculate the volume and the total surface area of a cone of base 16cm and slant height of 17cm
Need the vertical height, not slant height, but
h^2 + 8^2 = 17^2
h = √225 = 15
vol = (1/3)π r^2 h = (1/3)π(64)(15) = 320π cm^3
SA = circular base + surface of cone
= π(8^2) + π r l, where l is the slant height
= 64π + π(8)(17) = 200π cm^2
Calculate the volume and the total surface area of a cone of base diameter 16cm and slant height 17cm in terms of pie
To find the height of the cone, we need to use the Pythagorean theorem:
h^2 + 8^2 = 17^2
h^2 = 17^2 - 8^2
h^2 = 225
h = 15cm
The radius of the base is half the diameter, so r = 8cm.
The volume of the cone is:
V = (1/3)πr^2h
V = (1/3)π(8^2)(15)
V = 320π/3 cubic cm
To find the total surface area, we need to add the area of the base to the lateral surface area.
The area of the base is:
A_base = πr^2
A_base = π(8^2)
A_base = 64π square cm
The lateral surface area (or curved surface area) can be found using the slant height:
A_lateral = πrl
A_lateral = π(8)(17)
A_lateral = 136π square cm
The total surface area is:
A_total = A_base + A_lateral
A_total = 64π + 136π
A_total = 200π square cm
Why did the cone go to school? Because it wanted to become a "cone-puter" engineer! Now, let's calculate the volume and surface area of your cone.
To find the volume of a cone, we can use the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone.
Given that the base of the cone has a diameter of 16 cm, the radius is half of that, which is 8 cm. And we also know the slant height, which is 17 cm. However, we still need to find the height of the cone.
Using the Pythagorean theorem, we can calculate the height (h) of the cone. The slant height (17 cm) is the hypotenuse, and the radius (8 cm) is one of the legs. Applying the theorem, we have h² = 17² - 8².
Let's do the math. 17² equals 289 and 8² equals 64. So, h² = 289 - 64, which gives us h² = 225. Taking the square root of both sides, we find h = 15 cm.
Now we can calculate the volume using the formula. V = (1/3)π8²15. Plugging in the values, we have V = (1/3)π 480. The volume of the cone is approximately 502.65 cubic cm.
As for the surface area of the cone, we can use the formula A = πr(r + l), where r is the radius and l is the slant height.
Plugging in the given values, we have A = π8(8 + 17). Simplifying, A = π8(25), which is approximately 628.32 square cm.
So, the volume of the cone is approximately 502.65 cubic cm, and the total surface area is approximately 628.32 square cm. Don't worry, there's no need to cone-fuse yourself with the calculations!
To calculate the volume and the total surface area of a cone, we'll use the formulae:
Volume of a Cone = (1/3) * π * r^2 * h
Total Surface Area of a Cone = π * r (r + l)
where:
- r is the radius of the base of the cone
- h is the height of the cone
- l is the slant height of the cone
- π is a mathematical constant approximately equal to 3.14159
Now, let's calculate the volume and total surface area of the given cone step-by-step:
1. Find the radius (r):
The diameter of the base is 16 cm, so the radius (r) will be half of that.
r = 16 cm / 2 = 8 cm
2. Calculate the height (h) using the Pythagorean theorem:
The slant height (l) and the radius (r) form a right triangle with the height (h) as the third side.
Applying the Pythagorean theorem, we have:
h = √(l^2 - r^2)
h = √(17^2 - 8^2)
h = √(289 - 64)
h = √225
h = 15 cm
3. Calculate the volume of the cone:
Volume = (1/3) * π * r^2 * h
Volume = (1/3) * 3.14159 * 8^2 * 15
Volume ≈ 1005.31 cm³
4. Calculate the total surface area of the cone:
Total Surface Area = π * r (r + l)
Total Surface Area = 3.14159 * 8 (8 + 17)
Total Surface Area ≈ 685.94 cm²
So, the volume of the cone is approximately 1005.31 cm³ and the total surface area is approximately 685.94 cm².