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4.6 G) Cones – Part 1
4.6 G) Cones – Part 1
A cone is a 3D object that has a circular base and a smooth curved side that ends in a point (the point is known as an apex or vertex). An example of a cone is shown below:
We work out the volume of a cone by using the same formula that we used to work out the volume of a pyramid. The volume of a pyramid formula is:
A cone has a circular base. This means that we can modify the above formula because we can replace the area of base with the formula for working out the area of a circle. The formula for working out the area of a circle is shown below:
We sub in the area of a circle formula is for the area of the base. Also, I have let h be the perpendicular height of the cone.
Example 1
What is the volume of the cone below? Give your answer to one decimal place.
What is the volume of the cone below? Give your answer to one decimal place.
We will find the volume by using the volume of a cone formula. The formula is given below:
The above formula requires us to use the radius of the circle. The question gives us the diameter, which means that we need to divide the diameter by 2 to obtain the radius of the circle. Therefore, the radius is 10 cm (20 ÷ 2). Similar to the volume of a pyramid, h refers to the perpendicular height and the perpendicular height for this pyramid is 12 cm. Let’s sub these values into the above formula to obtain volume of the cone.
The final step is to round to one decimal place.
The volume of the cone is 1256.6 cm3 to one decimal place.
Surface Area of Cones
A cone has two faces; the base and the curved surface. The base of the cone is a circle, and we can work out the area of a circle by using the area of a circle formula:
A cone has two faces; the base and the curved surface. The base of the cone is a circle, and we can work out the area of a circle by using the area of a circle formula:
The formula for working out the area of the curved faced is:
r in the formula above in the radius and l is the slanted height. The slanted height is shown on the cone below.
It may be the case that you have to work out the slanted height by using Pythagoras’ theorem (we would need to do this if we were working out the surface area for the cone in the first example because we weren’t given the length of the slanted height. We will find the surface area for the cone in the first example in the next section).
We can combine the area of the base and the area of the curved surface to create a formula for working out the surface area of a cone:
We can combine the area of the base and the area of the curved surface to create a formula for working out the surface area of a cone:
Example 2
What is the surface area of the cone below? Give your answer to one decimal place.
What is the surface area of the cone below? Give your answer to one decimal place.
The radius of the cone is 4 cm and the slanted height is 10 cm. We can sub these values into the surface area of a cone formula.
We now need to round to one decimal place.
The total surface area of the cone is 175.9 cm2 to one decimal place.