A paper cone has base radius of 4cm and height of 3cm
a. Calculate the volume of the cone
b. Sketch the cone and hence find the slant height
c. Calculate the total surface area
a. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height. Plugging in the values given, we have V = (1/3)π(4^2)(3) = 16π cm^3.
b. To sketch the cone, draw a circle with a radius of 4 cm as the base. From the center of the base, draw a line perpendicular to the base that is 3 cm long to represent the height of the cone. Connect the endpoints of the line to the circumference of the base to form the curved surface of the cone. This curved surface is the slant height.
c. The total surface area of a cone is given by the formula A = πr(r + l), where r is the radius of the base and l is the slant height. Plugging in the values given, we have A = π(4)(4 + l). Since the slant height is equal to the height in a right circular cone, the slant height in this case is 3 cm. Plugging this value in, we have A = π(4)(4 + 3) = π(4)(7) = 28π cm^2.
a. To calculate the volume of the cone, we can use the formula:
Volume = (1/3) * pi * r^2 * h
where r is the base radius and h is the height.
Substituting the given values:
Volume = (1/3) * pi * 4^2 * 3
= (1/3) * pi * 16 * 3
= (1/3) * pi * 48
= 16 * pi
≈ 50.27 cm^3
Therefore, the volume of the cone is approximately 50.27 cm^3.
b. To sketch the cone and find the slant height, we can use the following steps:
- Draw a circle with a radius of 4 cm, which represents the base of the cone.
- From the center of the circle, draw a perpendicular line that represents the height of the cone. Label it as 3 cm.
- Connect the top point of the height line to the circumference of the circle, forming a triangle.
- The line connecting the center of the circle to the top point (height line) represents the slant height, denoted as "s".
c. To calculate the total surface area of the cone, we need to find the area of the base and the lateral surface area.
The area of the base (circle) can be found using the formula:
Base Area = pi * r^2
Substituting the given base radius:
Base Area = pi * 4^2
= pi * 16
= 16 * pi
The lateral surface area of the cone can be calculated using the formula:
Lateral Surface Area = pi * r * s
Substituting the values of the base radius and slant height:
Lateral Surface Area = pi * 4 * s
Now, to find the slant height, we can use the Pythagorean theorem:
s^2 = r^2 + h^2
s^2 = 4^2 + 3^2
s^2 = 16 + 9
s^2 = 25
s = 5 cm
Substituting the value of s:
Lateral Surface Area = pi * 4 * 5
= 20 * pi
The total surface area of the cone is the sum of the base area and the lateral surface area:
Total Surface Area = Base Area + Lateral Surface Area
= 16 * pi + 20 * pi
= 36 * pi
≈ 113.1 cm^2
Therefore, the total surface area of the cone is approximately 113.1 cm^2.