Find the volume of a cone,if the height is 12 cm and the base dianmeter is 7 cm
-calculate the slant height
-calculate the total surface area
Solve the question in the mathematical order
Area of a cone
just consult your formulas
v = 1/3 πr^2 h
now take a side view. Drop an altitude from the vertex to the center of the base. The slant height s can be found via
s^2 = h^2 + r^2
Now the area is the circular base plus the lateral area:
a = πr^2 + πrs
To find the volume of a cone, you'll need the formula:
Volume = (1/3) * π * r^2 * h
Where:
- r is the radius of the base of the cone
- h is the height of the cone
- π is a mathematical constant approximately equal to 3.14159
First, let's find the radius of the base of the cone. We are given the diameter of 7 cm. The radius (r) can be calculated by dividing the diameter by 2:
r = 7 cm / 2 = 3.5 cm
Now, we can substitute the values into the formula:
Volume = (1/3) * 3.14159 * (3.5 cm)^2 * 12 cm
≈ 3.14159 * 3.5 cm * 3.5 cm * 12 cm / 3
≈ 3.14159 * 12 cm * (3.5 cm)^2 / 3
≈ 3.14159 * 12 cm * 12.25 cm / 3
≈ 146.097 cm^3 (rounded to three decimal places)
Therefore, the volume of the cone is approximately 146.097 cm^3.
Now, let's calculate the slant height of the cone.
The slant height (l) of the cone can be found using the Pythagorean theorem. It is the hypotenuse of a right triangle, where the height (h) is one side and the radius (r) is the other side.
The Pythagorean theorem states that:
l^2 = r^2 + h^2
Substituting the given values:
l^2 = (3.5 cm)^2 + (12 cm)^2
l^2 ≈ 12.25 cm^2 + 144 cm^2
l^2 ≈ 156.25 cm^2 + 144 cm^2
l^2 ≈ 300.25 cm^2
Taking the square root of both sides:
l ≈ √300.25 cm^2
l ≈ 17.33 cm (rounded to two decimal places)
Therefore, the slant height of the cone is approximately 17.33 cm.
Next, let's calculate the total surface area of the cone.
The total surface area includes the area of the base and the curved surface area.
The area of the base (A_base) can be calculated using the formula for the area of a circle:
A_base = π * r^2
Substituting the given values:
A_base = 3.14159 * (3.5 cm)^2
A_base ≈ 3.14159 * 12.25 cm^2
A_base ≈ 38.4848 cm^2 (rounded to four decimal places)
The curved surface area (A_curved) can be calculated using the formula:
A_curved = π * r * l
Substituting the given values:
A_curved = 3.14159 * 3.5 cm * 17.33 cm
A_curved ≈ 3.14159 * 60.4555 cm^2
A_curved ≈ 190.9857 cm^2 (rounded to four decimal places)
Adding the area of the base and the curved surface area, we get the total surface area (A_total):
A_total = A_base + A_curved
A_total ≈ 38.4848 cm^2 + 190.9857 cm^2
A_total ≈ 229.4705 cm^2 (rounded to four decimal places)
Therefore, the total surface area of the cone is approximately 229.4705 cm^2.