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4.6 K) Composite Shapes
4.6 K) Composite Shapes
Composite shapes are created from simple 3D shapes. You may be given a composite shape and asked to work out what the volume or surface area of the shape is:
Let’s have a look at an example.
- To work out the volume of a composite shape, you work out the volume of the individual components and then add them together.
- To work out the surface area you need to work out the surface area of all of the faces that are on the outside and add all of these areas together.
Let’s have a look at an example.
Example 1
The 3D shape below is made by placing a hemisphere on top of a cone. What is the volume and surface area of the composite shape below? Give your answer to one decimal place.
The 3D shape below is made by placing a hemisphere on top of a cone. What is the volume and surface area of the composite shape below? Give your answer to one decimal place.
Volume
Let’s work out the volume first. The composite shape is made up of a cone and a hemisphere. We work out the volume of the composite shape by working out the volume of the individual shapes, and then we add them all together.
Let’s work out the volume first. The composite shape is made up of a cone and a hemisphere. We work out the volume of the composite shape by working out the volume of the individual shapes, and then we add them all together.
I am going to work out the volume of the cone first. The formula for working out the volume of a cone is:
r in the above formula stands for the radius and h stands for the perpendicular height. The radius for the cone is 5 cm and the perpendicular height is 12 cm. We sub these values into the formula to find the volume.
This is not the final answer, so we should not round.
I am now going to work out the volume of the hemisphere. A hemisphere is half a sphere. This means that we can calculate the volume of a sphere and then multiply by a half (or divide by 2) to obtain the volume of a hemisphere.
I am now going to work out the volume of the hemisphere. A hemisphere is half a sphere. This means that we can calculate the volume of a sphere and then multiply by a half (or divide by 2) to obtain the volume of a hemisphere.
The radius of the sphere is 5, so we sub r as 5 into the formula.
This is not the final answer, so we should not round.
The final step to calculate the volume of the composite shape is to add these two volumes together.
The final step to calculate the volume of the composite shape is to add these two volumes together.
We have the volume of the composite shape. The final step is to round our answer to one decimal place.
The volume of the composite shape to one decimal place is 576 cm3.
Surface Area
Now let’s work out what the surface area of the composite shape is. We calculate the surface area of a composite shape by adding the surface area of the faces that are on the outside of the composite shape. The composite shape in the example has two faces that are on the outside; the curved part of the cone and the curved part of the hemisphere. We are going to find the areas of each of these faces seperately and add them together.
I am going to find the area of the curved part of the cone first. The formula for working out the area of the curved part of a cone is:
Now let’s work out what the surface area of the composite shape is. We calculate the surface area of a composite shape by adding the surface area of the faces that are on the outside of the composite shape. The composite shape in the example has two faces that are on the outside; the curved part of the cone and the curved part of the hemisphere. We are going to find the areas of each of these faces seperately and add them together.
I am going to find the area of the curved part of the cone first. The formula for working out the area of the curved part of a cone is:
r in the above formula is the radius (which is 5 cm) and l is the slanted height (which is 13 cm). We sub r as 5 and l as 13 into the formula.
We now need to work out the curved surface area of the hemisphere. A hemisphere is half a sphere, and this means that the curved part of the hemisphere will be half of the surface area of a full sphere. This means that we obtain the curved surface area of a sphere by multiplying the surface area of a whole sphere by a half (or dividing by 2).
The radius for the hemisphere is 5 cm. Therefore, we sub r as 5 into the formula.
The final step is to add these surface areas together.
We have the surface area of the composite shape. The final step is to round our answer to one decimal place.
The total surface area of the composite shape is 361.3 cm2.