Surface area
How much cardboard do you need for a box? How much sheet metal for a tin? To find out, you need the surface area of the solid: the sum of the areas of all its faces. The trick is always the same: imagine the solid unfolded, lying flat on the table.
Lateral and total surface area
An artisan is making giant wooden dice for a games centre. If each die is a perfect cube with an edge length of $a = 12 \text{ cm}$, calculate the total surface area of…
An artisan is making giant wooden dice for a games centre. If each die is a perfect cube with an edge length of $a = 12 \text{ cm}$, calculate the total surface area of wood they will have to paint on each of the dice.
Calculate the total surface area of a shoebox in the shape of a cuboid whose dimensions are $15 \text{ cm}$ wide, $30 \text{ cm}$ long and $10 \text{ cm}$ high. Draw a…
Calculate the total surface area of a shoebox in the shape of a cuboid whose dimensions are $15 \text{ cm}$ wide, $30 \text{ cm}$ long and $10 \text{ cm}$ high. Draw a sketch of its net to help you with the calculation.
You can do it on paper and send a photo with your answer.
Surface area of a prism
The lateral faces of a right prism, when unfolded, form a rectangle whose base is the perimeter of the base of the prism and whose height is the height of the prism.
$$A_L = P_{base} \cdot h \qquad\qquad A_T = A_L + 2 \cdot A_{base}$$For a design and technology project, you need to cover a cylindrical pencil pot with adhesive paper. The pot has a base radius of $r = 4 \text{ cm}$ and a height of…
For a design and technology project, you need to cover a cylindrical pencil pot with adhesive paper. The pot has a base radius of $r = 4 \text{ cm}$ and a height of $h = 12 \text{ cm}$. Calculate how many square centimetres of paper you will need, knowing that the pot is open at the top (it only has one base). Take $\pi \approx 3,14$.
For a design and technology project, you have to make a wooden box in the shape of a regular hexagonal prism. The base edge measures $4\text{ cm}$ and the height of the…
For a design and technology project, you have to make a wooden box in the shape of a regular hexagonal prism. The base edge measures $4\text{ cm}$ and the height of the prism is $15\text{ cm}$. Calculate the total surface area of wood you will need to build the box (including the two bases).
(Note: To find the area of the base, remember that you must first calculate the apothem of the hexagon using Pythagoras' theorem).
Calculate the total surface area of a cube knowing that the diagonal of one of its faces is $10\text{ cm}$. (Hint: Use Pythagoras' theorem on one of the faces to find…
Calculate the total surface area of a cube knowing that the diagonal of one of its faces is $10\text{ cm}$.
(Hint: Use Pythagoras' theorem on one of the faces to find the edge length of the cube before calculating the area).
Surface area of a pyramid
The lateral faces of a regular pyramid are identical triangles, whose base is the side $l$ of the base and whose height is the slant height of the pyramid $ap$ (which is not the same as the height $h$).
$$A_L = \frac{P_{base} \cdot ap}{2} \qquad\qquad A_T = A_L + A_{base}$$A garden tent is shaped like a right regular triangular prism. The base is an equilateral triangle with side length $l = 3 \text{ m}$. If the length of the tent (height…
A garden tent is shaped like a right regular triangular prism. The base is an equilateral triangle with side length $l = 3 \text{ m}$. If the length of the tent (height of the prism) is $5 \text{ m}$, calculate the lateral area of the tent (excluding the two triangular bases).
In a designer ice cream parlour, they have created a wafer shaped like a right cone whose height is $12\text{ cm}$ and whose base radius is $5\text{ cm}$. Calculate the…
In a designer ice cream parlour, they have created a wafer shaped like a right cone whose height is $12\text{ cm}$ and whose base radius is $5\text{ cm}$. Calculate the total surface area of the wafer (curved surface area plus the base circle) to find out how much chocolate they need to coat it completely.
(Hint: First you will need to find the slant height of the cone).
Surface area of a cylinder and a cone
The cylinder and the cone unfold in the same way as polyhedra, but their lateral faces are curved:
$$A_{L,\text{cylinder}} = 2\pi r h \qquad A_{L,\text{cone}} = \pi r g$$An artisan candle maker wants to cover the lateral surface of a cylindrical candle with silver protective paper. The candle has a diameter of $8\text{ cm}$ and a height…
An artisan candle maker wants to cover the lateral surface of a cylindrical candle with silver protective paper. The candle has a diameter of $8\text{ cm}$ and a height of $20\text{ cm}$. If the paper only needs to cover the curved part (excluding the bases), what surface area of paper is needed for each candle? Use $\pi \approx 3,14$.
A decorative garden lighthouse is made up of two solids: a cylinder of radius $10\text{ cm}$ and height $30\text{ cm}$, topped by a cone with the same base and a slant…
A decorative garden lighthouse is made up of two solids: a cylinder of radius $10\text{ cm}$ and height $30\text{ cm}$, topped by a cone with the same base and a slant height of $15\text{ cm}$. Calculate the total outer surface area of this lighthouse (excluding the base on which it rests).
Surface area of a sphere
A sphere cannot be unfolded, but its surface area is found with a very simple formula: four times the area of its great circle.
$$A = 4\pi r^2$$Observe the net of this geometric solid. If each of the isosceles triangles has an area of $15 \text{ cm}^2$ and the central square has a side of $6 \text{ cm}$, what is…
Observe the net of this geometric solid. If each of the isosceles triangles has an area of $15 \text{ cm}^2$ and the central square has a side of $6 \text{ cm}$, what is the total area of the figure when assembled? What geometric solid is formed?
Calculate the curved surface area of a cylinder whose base circumference measures $C = 25,12 \text{ cm}$ and whose height is $h = 10 \text{ cm}$. (Hint: Remember that…
Calculate the curved surface area of a cylinder whose base circumference measures $C = 25,12 \text{ cm}$ and whose height is $h = 10 \text{ cm}$. (Hint: Remember that when you unfold the cylinder, the curved surface area is a rectangle whose base is the length of the circumference).
Back to the beginning: For the shoebox measuring $30 \times 20 \times 12$ cm you need $2\,400\text{ cm}^2$ of cardboard, and for the tin of preserves you need about $245\text{ cm}^2$ of sheet metal.