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mathematics UK Year 11 Mathematics (KS4)
 
Solved problems for the chapter:
 

Volume of 3D Shapes

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Surface area and volume of a sphere
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Cone for KS3
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Surface area and volume of a pyramid
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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:volume_of_3d_shapes/3807/


Example 1.

The lateral surface area of an upright cone is stretched in a plane. We obtain a circular section equal to half a circle with radius . Calculate the total surface area and volume of this upright cone.

    
Example 2.

Two spheres have radii in the ratio 2:3. What is the ratio of their volumes?

    
3.

The base area and the curved surface area of an equilateral cone measures . What is the radius of the cone?

    
4.

We need gold to coat a pendant in the shape of an equilateral cone. Calculate the volume of the pendant.

    
5.

In a cone with a diameter of we can pour liquid. Calculate the slant height of the cone.

    
6.

The area of the base area of an equilateral cone measures . Calculate the total surface area of the cone.

    
7.

How many wafers do we need to make a tall ice cream cone with an opening diameter ?

    
8.

The cone jacket measures . If the radius and the slant height of the cone are in the ratio , calculate the height of the cone.

    

9.

What cone has the area of the circular base and curved surface area in a ratio of ?

    
10.

The slant height of a cone is three times its radius and the height of the cone measures . Calculate the diameter of the cone.

    
11.

The height of a cone is four times its radius and the area of its curved surface measures . Calculate the volume of the cone.

    
12.

The perimeter of the triangle formed by the projection of a cone is . If the radius and the slant height are in the ratio of , calculate the curved surface area of the cone.

    
13.

The base area of a cylinder is and its curved surface area is . What is the total surface area of the cylinder?

    
14.

The cylindrical roller mesh has a total total surface area of . Its curved surface area measures . What is the area of the circular base?

    
15.

A cylindrical pipe with total surface area has a curved surface area of . What is the value of the base area?

    
16.

The base area of a cylinder of volume is . What is the height of the cylinder?

    
17.

The volume of a cylinder measures and the height . What is the area of the base of the cylinder?

    
18.

Calculate the total surface area of a cylinder with base area and height .

    

19.

A cylinder with height , has a circle with a diameter of as its base. Calculate the volume of the cylinder in cubic centimetres.

    
20.

The curved surface area of a cylinder is and the height . Calculate the total surface area of the cylinder.

    
21.

The curved surface area of an equilateral cylinder measures . Calculate the height of the cylinder.

    
22.

The curved surface area of a cylinder whose height is is . Calculate the diameter of the cylinder.

    
23.

The circumference of the base of an equilateral cylinder is measured . Calculate the total surface area and volume of the cylinder .

    
24.

If the roller is spread out on a flat surface, it takes up surfaces. What is the height of the cylinder if the area of the base measures ?

    
25.

A cylinder holds water. The height of the cylinder is . Calculate the total surface area of the cylinder.

    
26.

A hay bale with a total surface area of fits snugly in a cube-shaped packing box. What is the height the bale?

    
27.

The curved surface area of the cylinder is . What is the area of the base if the radius and height of the cylinder are in the ratio ?

    
28.

The base area and the casing of a cylinder with a diameter of are in the ratio . Calculate height of the cylinder.

    

29.

The curved surface area of a cylinder measures and the height is five times the diameter of the cylinder. Calculate the volume of the cylinder.

    
30.

If we cut an oak log, we get boards. The conveyor belt on the saw must be at least wide for cutting this log. What should be the minimum length of the trailer of the truck that brought the log from the forest? Assume no cutting losses .

    
31.

David has a rectangular garden with sides and long. He wants to extend the shorter side.


  • Write down the dependence of perimeter and area on the extended side.

  • How much does the new volume and width differ from the original if the shorter side is extended by ?

    
32.

Let's have a rectangle with a width b and length a that is units longer than the width.


  • In the form of a function rule, write the dependence of the side a on the width b.

  • Express the perimeter of the rectangle P with width b.

  • If we increase the width by , how much will the perimeter increase?

    
33.

What must be the minimum distance between the chassis of the vehicle and the carriageway at which a road cone of volume standing on of the carriageway can still be driven?

    
34.

A group of friends went on a trip by car. After of the regional road, they reached the highway, where they accelerated to a speed of .


  • Write down the function rule for calculating the total route on the regional road and highway.

  • Calculate how many kilometers they traveled in total after hours on the highway?

  • After hours of driving on the highway, they reached the exit and drove on the old road. How long did they drive on the old road if they reached their destination after from the start of their journey?

    
35.

The curved surface area and the base area of a cylinder are in a ratio of . What is the area of the curved surface if the radius of the cylinder is equal to ? Give the result obtained in square decimetres.

    
36.

The area of the base of a square base pyramid measures , and the lateral edge measures . Calculate the total surface area of the pyramid.

    
37.

We have three gold cubes with volumes , and . We melt the cubes and get a new, larger cube. What will be the length of the edge of the newly formed cube?

    
38.

We have three iron cubes with edges , and . We melt the cubes and pour into a mold to make a new larger cube from. What will be the volume of the newly formed cube?

    

39.

The shell of a regular triangular base prism with length of its base measures . What is the height of the prism?

    
40.

The surface area of the base of a cone with volume measures . What is the height of the cone?