We want to calculate the volume and then the lateral area of a cone. In order to do so, we should first calculate the radius from the given circumference
Radius
Let's recall the formula for the circumference.
C = 2 π r
We can substitute 28π for C and solve the equation above for r.
Now that we know the radius, we can calculate the volume and the surface area.
Volume
To find the volume, we can use the formula for the volume of a cone.
V= 13π r^2 h
Here, r is the radius and h is the height of the cone. We are given that the height of the cone is 18 inches and, earlier, we found that the radius of the base is 14 inches. Let's substitute these values into the formula and calculate the volume.
The volume of the cone is 1176π cubic inches. Now, let's move to calculating the lateral surface area.
Lateral Surface Area
To find the lateral surface area, we must know that the lateral area of a right cone is half the product of the circumference of the base and the slant height.
L.A.=1/2* 2π r* l ⇔ L.A.=π rl
In this formula, r is the radius and l is the slant height of the cone.
Although we do not have the slant height, we can find it using the Pythagorean Theorem.
In this right triangle, the legs are 14 inches and 18 inches long. The hypotenuse is l. We will substitute these values in the Pythagorean Theorem and solve for l.
The hypotenuse of the right triangle, and therefore the slant height of the cone, is sqrt(520) inches long. With this information we are able to calculate the lateral area of the cone. To do so, we will substitute r=14 and l=sqrt(520) into the formula for the lateral area. Let's do it!
