A cone has a base radius of 5cm and a height of 12cm calculate the total surface area
base: πr^2
lateral surface: πrs -- where s^2 = r^2 + h^2
Math
To calculate the total surface area of a cone, you need to find the sum of the lateral surface area and the base area. Here are the steps:
1. Calculate the lateral surface area:
- The lateral surface area of a cone can be calculated using the formula: π * r * l, where π is a mathematical constant approximately equal to 3.14159, r is the radius of the base, and l is the slant height.
- To find the slant height, you can use the Pythagorean theorem: l^2 = r^2 + h^2, where h is the height of the cone.
- Given that the radius (r) is 5 cm and the height (h) is 12 cm, we can calculate the slant height (l) using the formula: l = √(r^2 + h^2).
- Substitute the values into the lateral surface area formula: lateral surface area = π * r * l.
2. Calculate the base area:
- The base area of a cone can be calculated using the formula: π * r^2.
- Substitute the radius value into the formula: base area = π * r^2.
3. Calculate the total surface area:
- Total surface area = lateral surface area + base area.
Let's calculate step-by-step:
Step 1: Calculate the slant height
l = √(r^2 + h^2)
l = √(5^2 + 12^2)
l = √(25 + 144)
l = √169
l = 13 cm (rounded to two decimal places)
Step 2: Calculate the lateral surface area
lateral surface area = π * r * l
lateral surface area = 3.14159 * 5 * 13
lateral surface area ≈ 204.20350 cm^2 (rounded to two decimal places)
Step 3: Calculate the base area
base area = π * r^2
base area = 3.14159 * 5^2
base area ≈ 78.53975 cm^2 (rounded to two decimal places)
Step 4: Calculate the total surface area
total surface area = lateral surface area + base area
total surface area ≈ 204.20350 cm^2 + 78.53975 cm^2
total surface area ≈ 282.74325 cm^2 (rounded to two decimal places)
Therefore, the total surface area of the cone is approximately 282.74 cm^2.
To calculate the total surface area of a cone, you need to consider two parts: the curved surface area and the base area.
1. Curved Surface Area:
The curved surface area of a cone can be found using the formula: π * radius * slant height. The slant height of a cone can be calculated using the Pythagorean theorem: slant height = √(height^2 + radius^2).
So, for the curved surface area:
Curved Surface Area = π * radius * slant height
2. Base Area:
The base area of a cone is simply the area of a circle with radius r. Since the base of the cone is circular, the base area can be calculated using the formula: π * radius^2.
3. Total Surface Area:
The total surface area of the cone is the sum of the curved surface area and the base area.
Total Surface Area = Curved Surface Area + Base Area
Now let's calculate it step-by-step:
1. Calculate the slant height:
Slant height = √(height^2 + radius^2)
Slant height = √(12^2 + 5^2)
Slant height = √(144 + 25)
Slant height = √169
Slant height = 13 cm
2. Calculate the curved surface area:
Curved Surface Area = π * radius * slant height
Curved Surface Area = π * 5 cm * 13 cm
Curved Surface Area = 65π cm^2
3. Calculate the base area:
Base Area = π * radius^2
Base Area = π * 5 cm^2
Base Area = 25π cm^2
4. Calculate the total surface area:
Total Surface Area = Curved Surface Area + Base Area
Total Surface Area = 65π cm^2 + 25π cm^2
Total Surface Area = 90π cm^2
Therefore, the total surface area of the cone with a base radius of 5 cm and a height of 12 cm is 90π cm^2.