Exercises: Area

An artisan is making giant wooden dice for a toy library. If each die is a perfect cube with an edge of $a = 12 \text{ cm}$, calculate the total surface area of wood that will need to be painted on each of the dice.

Calculate the total area of a shoebox in the shape of a rectangular prism whose dimensions are $15 \text{ cm}$ wide, $30 \text{ cm}$ long and $10 \text{ cm}$ high. Draw a schematic of its net to help you with the calculation.

For a technology project, you need to cover a cylindrical pencil holder with sticky paper. The holder has a base radius of $r = 4 \text{ in}$ and a height of $h = 12 \text{ in}$. Calculate how many square inches of paper you will need, knowing that the holder is open at the top (it only has one base). Take $\pi \approx 3,14$.

A garden tent has the shape of a right regular triangular prism. The base is an equilateral triangle with a side of $l = 3 \text{ m}$ and a triangle height of $1,7 \text{ m}$. 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).

Find the total area of a regular square-based pyramid if we know that the side of the square base measures $8 \text{ cm}$ and the slant height of its lateral faces measures $10 \text{ cm}$.

A basketball has a radius of $r = 12 \text{ cm}$. Calculate the surface area of leather needed to make it, rounding the result to two decimal places. (Remember that the area of a sphere is $A = 4 \cdot \pi \cdot r^2$).

Look at the net of this geometric solid. If each of the equilateral 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 shape when it is assembled? What geometric solid is formed?

Calculate the lateral area of a cylinder whose base circumference measures $C = 25,12 \text{ cm}$ and its height is $h = 10 \text{ cm}$. (Hint: Remember that when you open the cylinder, the lateral area is a rectangle whose base is the length of the circumference).

For a 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 the Pythagorean theorem).

A tent for a youth event has the shape of a regular square-based pyramid. The side of the square base measures $10\text{ ft}$ and the height of the pyramid (from the center of the base to the top vertex) is $12\text{ ft}$. Calculate how many square feet of canvas were needed to make the four lateral faces of the tent.

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 must only cover the curved part (without the bases), what surface area of paper is needed for each candle? Use $\pi \approx 3,14$.

In a designer ice cream shop, they have created a wafer cone in the shape of a right cone whose height is $12\text{ cm}$ and its base radius is $5\text{ cm}$. Calculate the total area of the wafer cone (lateral surface plus the base circle) to find out how much chocolate they need to coat it completely.

(Hint: You will first need to find the slant height of the cone).

A modern planter has the shape of a frustum of a regular square-based pyramid. The bases are two squares with sides $20\text{ cm}$ and $12\text{ cm}$ respectively. If the height of the lateral faces (the slant height of the frustum) is $10\text{ cm}$, calculate the total outer surface area of the planter (including the top and bottom bases).

The dome of an astronomical observatory, which has the shape of a hemisphere with a radius of $6\text{ meters}$, is to be restored. If the budget for the paint is $15\text{ \$}$ per square foot, how much will it cost to paint the outer surface of the dome? (Use $\pi \approx 3,14$).

A decorative lighthouse for a garden is made up of two parts: a cylinder with a radius of $10\text{ cm}$ and a height of $30\text{ cm}$, 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 it rests on).

Calculate the total area of a cube knowing that the diagonal of one of its faces measures $10\text{ cm}$.

(Hint: Use the Pythagorean theorem on one of the faces to find the length of the cube's edge before calculating the area).

A design company wants to manufacture a ceiling lamp in the shape of a regular hexagonal pyramid. The side of the hexagonal base measures $10\text{ cm}$ and the height of the pyramid is $24\text{ cm}$. For the light to project correctly, they must cover the lateral faces with a special copper sheet. Calculate how many $\text{cm}^2$ of copper sheet are needed to cover the lateral surface of the lamp. Keep in mind that you will first need to find the apothem of the base and, with it, the apothem of the pyramid.

An architect has designed a skylight for a museum consisting of a frustum of a regular square pyramid made of glass. The bottom base has a side of $4\text{ m}$, the top base a side of $2\text{ m}$ and the height of the frustum is $1,5\text{ m}$. The glass used for the lateral faces costs $85\text{ \$/m}^2$. Calculate the budget needed to buy the glass for the lateral surface, rounding the result to two decimal places.

An acrylic trophy has been designed consisting of a right cylinder with a height of $10\text{ cm}$ and a diameter of $6\text{ cm}$, from which a perfect hemisphere has been extracted at the top base, whose radius coincides exactly with the radius of the cylinder. Calculate the total surface area of the trophy (outer cylindrical surface, bottom base surface and the curved surface of the hemispherical hollow). Use $\pi \approx 3,14$.

An industrial tank has the shape of a right cylinder topped by a cone with the same base. The height of the cylindrical part is $8\text{ m}$ and the height of the conical part is $3\text{ m}$. The diameter of the tank is $8\text{ m}$. It is desired to paint the entire outer lateral surface and the top base (cone), but not the base on which it rests. If each can of paint can cover $15\text{ m}^2$, what is the minimum number of cans of paint that must be bought?

A wooden cube with an edge of $12\text{ cm}$ is cut by a plane passing through the midpoints of three concurrent edges at the same vertex, separating a small triangular pyramid (an irregular tetrahedron). Calculate the total area of the resulting figure left after removing the small pyramid. Hint: The cut generates a new face which is an equilateral triangle whose sides are the diagonals of the squares with side $6\text{ cm}$.

A manufacturer of gym balls produces rubber spheres in two sizes. The small ball has a surface area of $5024\text{ cm}^2$. The large ball has a radius that is exactly double that of the small ball. Calculate the surface area of the large ball. What numerical relationship (ratio) is there between the area of the large ball and that of the small one? Use $\pi = 3,14$.

A greenhouse is to be built in the shape of a half-cylinder resting on the ground, as shown in the figure. The length of the greenhouse is $15\text{ m}$ and the width of the base (the diameter of the semi-cylinder) is $6\text{ m}$. If the plastic to cover the curved part and the two front walls (semicircles) costs $12\text{ \$/m}^2$, what will be the total cost of the plastic needed? (The floor is not counted).

A mechanical steel part has the shape of a right square-based prism with a cylindrical hole passing through it from base to base through the center. The side of the square base of the prism measures $10\text{ cm}$ and the height of the part is $20\text{ cm}$. The diameter of the cylindrical hole is $8\text{ cm}$. Calculate the total surface area of the part (including the inner surface of the hole).