Surface area and volume

A hemisphere with radius $r$ cm is attached to a cone of radius $r$ cm and height $h$ cm to make the solid. The hemisphere's volume is one third of the volume of the whole solid.

May/June 2015

A solid is made by attaching a cone of radius $4.5\text{ cm}$ and height $7.6\text{ cm}$ to a hemisphere of radius $4.5\text{ cm}$.

May/June 2015

For a sphere, the volume is $\frac{4}{3}\pi r^3$. For a cone, the volume is $\frac{1}{3}\pi r^2 h$. A cone is taken away from a solid wooden hemisphere with radius $3\text{ cm}$. The cone also has radius $3\text{ cm}$ and height $2\text{ cm}$. The volume of the wood left is $k\pi \text{ cm}^3$.

May/June 2016

Solid I takes the form of a cylinder with a smaller cylinder cut out from the middle.

May/June 2016

A pyramid’s volume = $\frac{1}{3} \times$ base area $\times$ perpendicular height. The diagrams depict a solid pyramid $L$ split into two sections, $M$ and $N$, by a plane parallel to the base. Pyramid $L$ has a rectangular base measuring $9\text{ cm}$ by $12\text{ cm}$. Its perpendicular height is $30\text{ cm}$.

May/June 2017

This container is formed from thin material and has the shape of an open-topped cuboid. Its length measures $15\text{ cm}$ and its width measures $8\text{ cm}$. The container's volume is $720\text{ cm}^3$.

May/June 2017

A paving slab is a cuboid whose length is $40\text{ cm}$, width is $20\text{ cm}$ and depth is $h\text{ cm}$. The volume of the slab is $2400\text{ cm}^3$.

May/June 2018

The diagram represents a pyramid. The square base, $ABCD$, has an edge length of $3$ cm. The base lies horizontally, and vertex $E$ is directly above $D$, where $ED=4$ cm.

May/June 2018

The diagram represents a triangular prism that is $12$ cm long. Its cross-section is a right-angled triangle with sides of $6$ cm, $8$ cm and $10$ cm. Using the grid, construct a net for this prism. Apply a scale where $1$ cm stands for $2$ cm. One face has already been drawn for you.

May/June 2019

A cuboid has a square-shaped base. The base length of the cuboid is $y$ cm. Its height is double the length of the base. The cuboid’s total surface area is $360\,\text{cm}^2$.

May/June 2019

The cuboid has a square base with side length $4\text{ cm}$. Its volume is $48\text{ cm}^3$.

May/June 2021

The combined volume of two cubes is $152\text{ cm}^3$. One cube has an edge length of $5\text{ cm}$.

May/June 2022

The diagram presents the net of a solid on a $1\text{ cm}$ grid.

May/June 2022

A cuboid measures $x\text{ cm}$ by $x\text{ cm}$ by $10\text{ cm}$. Its volume is $62.5\text{ cm}^3$. Determine the value of $x$.

May/June 2022

The diagram depicts a pentagon. Every length is given in centimetres.

May/June 2022

A triangular prism is shown in the diagram. Its cross-section is a right-angled isosceles triangle.

May/June 2023

Determine the length and width of rectangle $R$.

May/June 2023

A cuboid measures $5\,\text{cm}$ by $12\,\text{cm}$ by $h\,\text{cm}$. Its volume is $480\,\text{cm}^3$. Calculate $h$.

May/June 2024

A cone is placed on top of a hemisphere to make a solid. The cone and the hemisphere each have radius $r$ cm. The total height of the solid is $10$ cm. The curved surface area of the hemisphere is $145\text{ cm}^2$.

May/June 2024

The area of a single face of the cube is $9\text{ cm}^2$.

May/June 2025

The diagram shows a solid made by attaching a cone to a hemisphere. The cone has diameter $14\text{ cm}$ and the hemisphere has diameter $14\text{ cm}$. The solid has a total height of $24\text{ cm}$.

May/June 2025

The diagram depicts a solid triangular prism. Every length is measured in centimetres.

Oct/Nov 2015

The diagram depicts a solid cone with radius $r$ cm, height $h$ cm and slant height $l$ cm. Each cone’s slant height is $4$ cm greater than its radius. Use $\pi = 3$ throughout this question.

Oct/Nov 2015

This cookery measuring spoon is made up of a hemispherical bowl and a handle. The bowl has an internal volume of $20\text{ cm}^3$, and the handle measures $5\text{ cm}$ in length.

Oct/Nov 2015

The volume of a sphere is $\frac{4}{3}\pi r^3$. In a storm, raindrops collect inside a cylinder that is standing on level ground. Before the storm began, the cylinder contained no water. The cylinder has radius $20\text{ mm}$. Each raindrop is a sphere with radius $2\text{ mm}$. When the storm has ended, the water depth in the cylinder is $16\text{ mm}$.

Oct/Nov 2016

For a sphere, the volume is $\frac{4}{3}\pi r^3$. For a sphere, the surface area is $4\pi r^2$. A hemispherical bowl is formed from material with thickness $0.8\text{ cm}$. The external rim radius of the bowl is $9\text{ cm}$. The bowl is fixed onto a base that is a solid cylinder, with radius $3.8\text{ cm}$ and height $1.5\text{ cm}$.

Oct/Nov 2016

The ventilation shaft has a cylindrical shape, with radius $0.4\,\text{m}$ and length $15\,\text{m}$. Calculate the volume of the cylinder.

Oct/Nov 2017

The diagram represents a closed box. It is a cuboid. The measurements are given in centimetres.

Oct/Nov 2018

The diagram represents a triangular prism. All dimensions are given in centimetres.

Oct/Nov 2018

The diagram depicts lamp $A$. It takes the form of a cylinder with a hemisphere placed on top. The radius of the hemisphere and the radius of the cylinder are both $3\text{ cm}$. The lamp’s total height is $24\text{ cm}$.

Oct/Nov 2018

This diagram presents the net of a solid.

Oct/Nov 2019

A tap lets water fall drop by drop into a container resting on a horizontal surface. The container is a cuboid whose base measures $5\text{ cm}$ by $4\text{ cm}$. The volume of one drop of water is $0.08\text{ cm}^3$.

Oct/Nov 2019

[Volume of cone $= \frac{1}{3}\pi r^2 h$] [Curved surface area of a cone $= \pi r l$] The sketch shows a gate post. It is formed as a cylinder with a cone placed on top. The cylinder and the cone each have diameter $8\,\text{cm}$. The cylinder has height $95\,\text{cm}$ and the cone has height $15\,\text{cm}$.

Oct/Nov 2019

The diagram presents a bowl with a circular base. Its curved outer face is produced by cutting away a cone of radius $12$ cm and height $45$ cm from a larger cone, as the diagram indicates. The radius across the top of the bowl is $16$ cm and its height is $15$ cm.

Oct/Nov 2019

Write the letter for each drawing that is the net of a cube.

Oct/Nov 2020

This net is then folded to form a triangular prism.

Oct/Nov 2020

The diagram depicts a garden shed standing on level ground. It has the form of a prism with trapezium $ABCD$ as its cross-section. The shed base, $ABFE$, is rectangular. $AB = 1.55$ m, $AD = 2.25$ m, $BC = 1.85$ m and $BF = 2.10$ m.

Oct/Nov 2020

Points A, B, C and D lie on the circle with centre $O$. $\angle BAD = 68^\circ$ and $\angle CBO = 52^\circ$.

Oct/Nov 2020

Solids, together with volume, surface area, and bounds.

Oct/Nov 2020

Write down the name of the solid that each net makes.

Oct/Nov 2021

Volume of a cone $= \dfrac{1}{3}\pi r^2 h$, curved surface area of a cone $= \pi r l$. Surface area of a sphere $= 4\pi r^2$. A solid cone has radius $y\,\text{cm}$. The slant height of the cone is $25\%$ greater than its radius. A solid sphere has radius $R\,\text{cm}$. The sphere's surface area is the same as the cone's total surface area.

Oct/Nov 2022

The diagram depicts a regular pentagon $ABCDE$ with centre $O$.

Oct/Nov 2022

A triangular prism is shown in the diagram. Calculate the volume of the prism. State the units for your answer.

Oct/Nov 2023

Volume of a cone $= \frac{1}{3}\pi r^2 h$. Curved surface area of a cone $= \pi r l$. A solid is made by removing a smaller cone from the middle of a larger cone. The small cone is mathematically similar to the large cone. The vertex of the large cone lies vertically below the vertex of the small cone. The height of the large cone is $21$ cm and the diameter of the top is $18$ cm. The height of the small cone is $14$ cm.

Oct/Nov 2023

The diagram depicts a sphere enclosed by a cube. The sphere is touching all 6 faces of the cube. The cube’s volume is $343\text{ cm}^3$. Calculate the volume of the sphere. (Volume of sphere $= \frac{4}{3}\pi r^3$)

Oct/Nov 2024

The diagram represents a tank. It is a cuboid with length $1.2\,\text{m}$, width $0.6\,\text{m}$ and height $h\,\text{m}$. Its volume is $1.8\,\text{m}^3$.

Oct/Nov 2024

The illustration shows a container that is empty. It is a cuboid with width 10 cm, length 8 cm and height 9 cm. Water enters the container by dripping at a rate of 6 millilitres per second.

Oct/Nov 2025

The diagram depicts a solid cylinder and a solid hemisphere. The cylinder's radius is the same as the hemisphere's radius. Their volumes are equal. The cylinder has a height of 6 cm.

Oct/Nov 2025

For this question, every length is measured in centimetres. The diagram shows the net of a cuboid with dimensions 3, 5.2 and 4.7.

Oct/Nov 2025