An open cone has a radius of 6 cm and a height of 8 cm. What is its surface area in square centimetre
The surface area of an open cone can be found by adding the lateral area (just the curved part) to the area of the base.
The formula for the lateral area of a cone is:
Lateral area = πrℓ
where r is the radius of the base and ℓ is the slant height, which can be found using the Pythagorean theorem:
ℓ^2 = r^2 + h^2
where h is the height of the cone.
Plugging in the values given, we have:
ℓ^2 = 6^2 + 8^2 = 100
ℓ = √100 = 10
So the lateral area of the cone is:
Lateral area = π(6)(10) = 60π cm^2
The area of the base is simply the area of a circle with radius 6:
Base area = π(6)^2 = 36π cm^2
Adding these together, we get the surface area of the cone:
Surface area = Lateral area + Base area = 60π + 36π = 96π ≈ 301.59 cm^2
Therefore, the surface area of the open cone is approximately 301.59 square centimetres.
60π
is approximately 188.5.
So the surface area of the open cone is approximately 188.5 + 36π, which is approximately 188.5 + 113.1, which equals 301.6 square centimetres (rounded to one decimal place).
To calculate the surface area of an open cone, we need to find the lateral surface area (LSA) and add it to the base area.
The LSA of a cone can be calculated using the formula:
LSA = π * r * l
Where:
π is a mathematical constant approximately equal to 3.14159
r is the radius of the base of the cone
l is the slant height of the cone
To find the slant height, we can use the Pythagorean theorem:
l^2 = r^2 + h^2
Where:
h is the height of the cone
Given:
Radius (r) = 6 cm
Height (h) = 8 cm
Calculating the slant height:
l^2 = r^2 + h^2
l^2 = 6^2 + 8^2
l^2 = 36 + 64
l^2 = 100
l = √100
l = 10 cm
Calculating the LSA:
LSA = π * r * l
LSA = 3.14159 * 6 * 10
LSA = 188.4956 cm^2 (rounded to 4 decimal places)
The base area of a cone can be calculated using the formula:
Base Area = π * r^2
Calculating the base area:
Base Area = π * 6^2
Base Area = 3.14159 * 36
Base Area = 113.0976 cm^2 (rounded to 4 decimal places)
To find the total surface area:
Total Surface Area = LSA + Base Area
Total Surface Area = 188.4956 + 113.0976
Total Surface Area = 301.5932 cm^2 (rounded to 4 decimal places)
Therefore, the surface area of the given open cone is approximately 301.5932 square centimeters.