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IB Mathematics: Applications and Interpretation HL · 12.10: Geometry & Trigonometry · 12.10.9

Name Date

Volumes and surface areas of solids Problem set A

Sketch a cross-section when the solid is made of more than one part. Give answers to 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. A cone has base radius 5.4 cm and perpendicular height 10.2 cm.

    10.2 cm5.4 cm
    1. (a) Find the slant height.
    2. (b) Find the volume.

  2. A cone has base radius 7 cm and slant height 18.5 cm.

    1. (a) Find the curved surface area.
    2. (b) Find the total surface area, including the base.

  3. A sphere has radius 8.5 cm.

    1. (a) Find its volume.
    2. (b) Find its surface area.
    3. (c) A solid hemisphere has the same radius. Find its total surface area, including the flat face.

  4. A sphere has volume 2500 cm³.

    1. (a) Find the radius of the sphere.
    2. (b) A cone has volume 330 cm³ and base radius 5 cm. Find its perpendicular height.

In context

  1. A hemispherical bowl has internal diameter 36 cm.

    1. (a) Find the capacity in litres.
    2. (b) How many 0.25 litre cups can be filled completely from a full bowl?

  2. A water tower holds its water in a cylinder of radius 3.2 m and height 7.5 m, with a hemispherical top of the same radius.

    6.4 m7.5 m
    1. (a) Find the volume.
    2. (b) Find the area of its curved outside surface (wall and top, not the floor).

  3. A garden tower is a cuboid with a square base of side 2.8 m and height 4.5 m, with a square-based pyramid of height 1.5 m on top, with the same base.

    2.8 m4.5 m
    1. (a) Find the total volume.
    2. (b) Find the total area of the four sloping triangular faces.

  4. A solid wax sphere of radius 7.5 cm is melted and recast into 15 identical cones. Each cone has base radius 3 cm. No wax is lost. Find the perpendicular height of each cone.

Stretch

  1. A toy is a solid hemisphere of radius 5 cm fixed to the base of a cone of the same base radius. The volume of the whole toy is 450 cm³. Find the perpendicular height of the cone.

  2. A conical filter has internal radius 6 cm at the top and internal depth 15 cm. It stands on its point and holds water to a depth of 10 cm.

    12 cm10 cm
    1. (a) Find the radius of the liquid surface.
    2. (b) Find the volume of liquid.
    3. (c) Find the volume of liquid as a percentage of the volume of the full container.

jmaths.xyz · AI HL · 12.10.9 Volumes and surface areas of solids