find the surface area of a cylinder with a radius of 4 cm and a height of 2 cm. use the approximation of 3.14 for pi
To find the surface area of a cylinder, you need to calculate the area of the two circular bases and the area of the curved surface (the lateral area).
The area of one circular base is given by the formula A = πr², where r is the radius. Since the cylinder has two bases, you multiply the area of one base by 2.
Given r = 4 cm, the area of one base is:
A_base = πr² = 3.14 * (4 cm)² = 3.14 * 16 cm² = 50.24 cm²
The area of two bases is:
2 * A_base = 2 * 50.24 cm² = 100.48 cm²
The lateral surface area of a cylinder is given by the formula A_lateral = 2πrh, where h is the height.
Given h = 2 cm, the lateral surface area is:
A_lateral = 2πrh = 2 * 3.14 * 4 cm * 2 cm = 50.24 cm²
Now, add the area of the two bases to the lateral surface area to get the total surface area:
Surface Area_total = Area_bases + A_lateral = 100.48 cm² + 50.24 cm² = 150.72 cm²
So, the surface area of the cylinder is 150.72 cm².
The formula for the total surface area of a cylinder is given by the sum of the areas of the circular bases and the lateral area (the area of the side). The formula is:
Total Surface Area (A_total) = 2 * Area of Base + Lateral Surface Area
Total Surface Area (A_total) = 2πr² + 2πrh
Where:
- π (Pi) is the constant approximately equal to 3.14.
- r is the radius of the circular base of the cylinder.
- h is the height of the cylinder.
Intuitively, the first term (2πr²) represents the area of the top and bottom circles, and the second term (2πrh) represents the area of the side of the cylinder when it is unwrapped into a rectangle.
First, let's find the radius of the cylinder, which is half of the diameter. The diameter is 12 inches, so the radius (r) is 12 inches / 2 = 6 inches.
Now, we'll use the formula for the total surface area of a cylinder:
Total Surface Area (A_total) = 2πr² + 2πrh
Let's plug in the values:
r = 6 inches
h = 10 inches
π ≈ 3.14
Total Surface Area (A_total) = 2 * 3.14 * (6 inches)² + 2 * 3.14 * 6 inches * 10 inches
First, we calculate the area of the top and bottom circles (2πr²):
Circle Area = πr² = 3.14 * (6 inches)² = 3.14 * 36 inches² = 113.04 inches²
Since there are two bases (top and bottom):
2 * Circle Area = 2 * 113.04 inches² = 226.08 inches²
Now, we calculate the lateral surface area (2πrh):
Lateral Surface Area = 2 * π * r * h = 2 * 3.14 * 6 inches * 10 inches = 376.8 inches²
Finally, we sum both areas to get the total surface area:
Total Surface Area (A_total) = 226.08 inches² + 376.8 inches² = 602.88 inches²
The surface area of the cylinder with a diameter of 12 inches and a height of 10 inches, using π ≈ 3.14, is 602.88 square inches.
To cover the cake with icing, the baker needs to calculate the surface area of the sides (lateral surface area) and the top of the cylinder cake. The bottom does not need icing.
We can use the formulas for the areas of the circular top and the lateral surface:
Top Area (A_top) = πr²
Lateral Surface Area (A_lateral) = 2πrh
Where:
π (Pi) is approximately 3.14,
r is the radius of the cylinder cake, which is 5 inches,
h is the height of the cylinder cake, which is 7 inches.
Let's start by calculating the area of the top:
A_top = πr² = 3.14 * (5 inches)² = 3.14 * 25 inches² = 78.5 inches²
Now, we calculate the lateral surface area:
A_lateral = 2πrh = 2 * 3.14 * 5 inches * 7 inches = 2 * 3.14 * 35 inches² = 219.8 inches²
The total amount of icing needed is the sum of the top area and the lateral surface area:
Total Icing Area = A_top + A_lateral = 78.5 inches² + 219.8 inches² = 298.3 inches²
Rounding to the nearest tenth, the baker will need 298.3 square inches of icing to cover the sides and top of the cake.
To refinish the side of the silo, the farmer needs to calculate the lateral surface area of the cylinder, as the top and bottom are not being refinished.
The formula for the lateral surface area (A_lateral) of a cylinder is given by:
A_lateral = 2πrh
Where:
- π (Pi) is approximately 3.14,
- r is the radius of the cylinder,
- h is the height of the cylinder.
Given the radius (r) of 4 feet and the height (h) of 30 feet, we can calculate the lateral surface area as follows:
A_lateral = 2 * π * r * h
A_lateral = 2 * 3.14 * 4 feet * 30 feet
A_lateral = 2 * 3.14 * 120 feet²
A_lateral = 2 * 376.8 feet²
A_lateral = 753.6 feet²
Rounding to the nearest tenth, the farmer will need 753.6 square feet of aluminum to refinish the side of the silo.