Calculate the total surface area of a solid cone of slant height of 25cm and vertical height of 24cm
To calculate the total surface area of a solid cone, we need to find the area of the curved surface (lateral surface) as well as the base.
The curved surface area (CSA) of a cone can be calculated using the formula:
CSA = π × r × l
where r is the radius of the base of the cone and l is the slant height.
We are given the slant height (l) to be 25 cm, and the vertical height (h) to be 24 cm. To find the radius (r) of the base, we can use the Pythagorean theorem:
r = √(l^2 - h^2)
= √(25^2 - 24^2)
= √(625 - 576)
= √49
= 7 cm
Now, we can calculate the curved surface area:
CSA = π × r × l
= π × 7 × 25
= 175π cm²
To find the base area, we can use the formula for the area of a circle:
Base area = π × r^2
= π × 7^2
= 49π cm²
The total surface area (TSA) of the solid cone is the sum of the curved surface area and the base area:
TSA = CSA + Base area
= 175π + 49π
= 224π cm²
Therefore, the total surface area of the solid cone is 224π square centimeters.
To calculate the total surface area of a cone, we need to find the area of the curved surface, also known as the lateral surface area, as well as the area of the base.
Step 1: Find the base radius.
The vertical height and slant height are given, and we can use them to find the base radius using the Pythagorean Theorem.
Let's call the base radius "r" and the slant height "L."
Using the Pythagorean Theorem: L^2 = r^2 + h^2
Substituting the given values: 25^2 = r^2 + 24^2
625 = r^2 + 576
r^2 = 625 - 576
r^2 = 49
r = √49
r = 7 cm
Step 2: Calculate the lateral surface area.
The lateral surface area of a cone is given by the formula LSA = π * r * L, where r is the base radius and L is the slant height.
Substituting the values: LSA = π * 7 cm * 25 cm
LSA = 175π cm^2
Step 3: Calculate the base area.
The base area of a cone is given by the formula BA = π * r^2, where r is the base radius.
Substituting the value: BA = π * (7 cm)^2
BA = 49π cm^2
Step 4: Add the lateral surface area and base area to get the total surface area.
Total surface area = LSA + BA
Total surface area = 175π cm^2 + 49π cm^2
Total surface area = 224π cm^2
So, the total surface area of the given cone is 224π square centimeters.
To calculate the total surface area of a solid cone, we need to consider the curved surface area (lateral surface area) and the base area.
The curved surface area of a cone can be calculated using the formula:
Curved Surface Area = π * r * l
Where:
- π (Pi) is a mathematical constant approximately equal to 3.14159
- r is the radius of the base
- l is the slant height of the cone
To find the radius (r) of the base, we can use the Pythagorean theorem. In a right triangle formed by the height (h), slant height (l), and radius (r) of the cone, we have:
l^2 = r^2 + h^2
Rearranging the equation, we get:
r = √(l^2 - h^2)
Now, let's substitute the given values into the formulas and calculate the total surface area.
Given:
Slant height (l) = 25 cm
Vertical height (h) = 24 cm
First, calculate the radius (r):
r = √(25^2 - 24^2)
r = √(625 - 576)
r = √49
r = 7 cm
Next, calculate the curved surface area:
Curved Surface Area = π * r * l
Curved Surface Area = 3.14159 * 7 * 25
Curved Surface Area ≈ 549.78 cm^2 (approx.)
Finally, calculate the base area:
Base Area = π * r^2
Base Area = 3.14159 * 7^2
Base Area ≈ 153.94 cm^2 (approx.)
To get the total surface area of the cone, add the curved surface area and the base area:
Total Surface Area = Curved Surface Area + Base Area
Total Surface Area ≈ 549.78 + 153.94
Total Surface Area ≈ 703.72 cm^2 (approx.)
Therefore, the total surface area of the solid cone is approximately 703.72 square centimeters.