Exercises: 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 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 sketch of its net to help you with the calculation.

You can do it on paper and send a photo with your answer.

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 $\pi \approx 3.14$.

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).

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

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

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 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 when you unfold the cylinder, the curved surface area is a rectangle whose base is the length of the circumference).

Cian wants to line a cylindrical pencil holder with adhesive paper. The holder is open at the top, its circular base has a radius of $4\text{ cm}$, and its height is $12\text{ cm}$. To calculate the amount of paper needed, Cian writes in his copy:\n- Lateral area: $A_L = 2 \times \pi \times 4 \times 12 \approx 301.59\text{ cm}^2$\n- Base area: $A_{\text{base}} = \pi \times 4^2 \approx 50.27\text{ cm}^2$\n- Total area: $A_T = A_L + 2 \times A_{\text{base}} = 301.59 + 2 \times 50.27 = 402.13\text{ cm}^2$\na) Point out the geometric reasoning error Cian made, considering the function of the object he is going to line.\nb) Explain what modification must be made to the total area formula based on the net of a cylinder that is open at the top.\nc) Calculate the actual surface area of adhesive paper that Cian needs to cut out (use $\pi \approx 3.14$).

For school training sessions of the Irish Team Handball Association, official spherical size 2 handballs are used, with a radius of $8.5\text{ cm}$.\na) Before performing any exact calculation, make a rough estimate of the spherical surface area by mentally rounding the radius to $8\text{ cm}$ and $\pi$ to $3$. Reason whether you think the actual area will exceed $1,000\text{ cm}^2$ or will be smaller.\nb) Now calculate the leather surface area of the ball precisely by applying the sphere area formula ($A = 4 \times \pi \times r^2$), using the actual radius of $8.5\text{ cm}$ and $\pi \approx 3.14$. Round the result to the nearest hundredth.\nc) Compare the value obtained with your initial estimate and assess why estimating a result beforehand helps us check whether calculator calculations are reasonable.

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).

A tent for a youth event is in the shape of a regular square pyramid. The side of the square base measures $10\text{ m}$ and the height of the pyramid (from the centre of the base to the top vertex) is $12\text{ m}$. Calculate how many square metres of canvas were needed to make the four side 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 only needs to cover the curved part (excluding the bases), what surface area of paper is needed for each candle? Use $\pi \approx 3.14$.

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).

A modern planter is shaped like a frustum of a regular square pyramid. The bases are two squares with sides $20\text{ cm}$ and $12\text{ cm}$ respectively. If the height of the side faces (the slant height of the frustum) is $10\text{ cm}$, calculate the total outer surface area of the planter (including upper and lower bases).

The dome of an astronomical observatory in the shape of a hemisphere of radius $6\text{ metres}$ is to be restored. If the budget for painting is $\text{€}\ 15$ per square metre, how much will it cost to paint the outer surface of the dome? (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 height of $15\text{ cm}$. Calculate the total outer surface area of this lighthouse (excluding the base on which it rests).

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).

A canned vegetable factory in Navarra produces cylindrical tin cans to package asparagus. The technical team has determined that the circular base of the can must have a radius of $4\text{ cm}$. The cutting machine has exactly $351.68\text{ cm}^2$ of sheet metal available to manufacture each complete closed can (lateral area plus the two circular lids).\na) Set up an equation where the unknown is the height $h$ of the cylinder, using the data provided and the total area formula ($A_T = 2 \times \pi \times r \times h + 2 \times \pi \times r^2$, using $\pi \approx 3.14$).\nb) Solve the equation to calculate the exact height the can must have.\nc) If the quality department indicates that for that amount of preserves the ideal height must be between $9\text{ cm}$ and $11\text{ cm}$, determine with justification whether the dimensions found meet that quality standard.

In the region of Connemara, County Galway, the conical roof of a windmill is going to be restored to protect its inner machinery from torrential rains recorded by Met Éireann. The roof is shaped like a regular cone: the radius of its circular base measures $3\text{ m}$ and the vertical height from the center of the base to the apex measures $4\text{ m}$. Only the outer lateral surface will be waterproofed (the cone has no base to cover, as it rests open on the masonry tower).\na) Model the cross-section of the roof using a right triangle to calculate the slant height of the cone by applying the Pythagorean theorem.\nb) Calculate the square meters of conical surface that must be waterproofed (take $\pi \approx 3.14$).\nc) To install the waterproofing sheets, an extra $10\,\%$ of material must be ordered due to cuts and overlaps. If one square meter of sheet costs €25, calculate the total cost of the material needed for the restoration.

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

An architect has designed a glass skylight for a museum consisting of a frustum of a regular square pyramid. The lower base has a side length of $4\text{ m}$, the upper base a side length of $2\text{ m}$ and the height of the frustum is $1.5\text{ m}$. The glass used for the side faces costs $85\text{ €/m}^2$. Calculate the budget required to buy the glass for the lateral surface area, rounding the result to two decimal places.

An acrylic trophy has been designed consisting of a right cylinder of height $10\text{ cm}$ and diameter $6\text{ cm}$, from which a perfect hemisphere has been removed at the upper base, whose radius matches exactly the radius of the cylinder. Calculate the total surface area of the trophy (outer cylindrical surface, surface of the bottom base and the curved surface of the hemispherical hollow). Use $\pi \approx 3.14$.

An industrial storage tank is shaped like 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 surface (cone), but not the base on which it rests. If each tin of paint can cover $15\text{ m}^2$, what is the minimum number of tins of paint that must be bought?

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

A gym ball manufacturer 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 is the numerical ratio 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 diagram. The length of the greenhouse is $15\text{ m}$ and the width of the base (the diameter of the half-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 required? (The floor is not included).

A mechanical steel part has the shape of a right square-based prism with a cylindrical hole running through it from base to base at the centre. 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).

To design a commemorative monolith in a public park, the local council plans a tower consisting of a right square prism with a base side of $4\text{ m}$ and a height of $10\text{ m}$, topped by a right square pyramid with an identical base and a vertical height of $1.5\text{ m}$. Two architects discuss how to calculate the total exterior surface area that will require water-repellent treatment (including the bottom base of the prism that rests on the prepared ground):\n- Aoife's method: "I calculate the total area of the complete pyramid using its formula, add the total area of the complete prism using its formula, and subtract twice the area of the square where they join."\n- Seán's method: "I look at the exterior net of the combined solid: I directly calculate the area of the bottom base, the lateral area of the prism, and the lateral area of the pyramid, adding those three quantities."\na) Formally prove that both methods arrive at the same algebraic expression for the surface area.\nb) Justify which of the two methods is more convenient and efficient to avoid errors in workshop practice.\nc) Apply the most convenient method to find the total exterior surface area to be treated. (Keep in mind that for the pyramid you must first calculate its slant height using the Pythagorean theorem with half of the base side and its vertical height).

An extra virgin olive oil cooperative in Kilkenny wants to market a premium designer metal can in the shape of a regular hexagonal prism. For ergonomic and kitchen-grip reasons, the edge of the hexagonal base measures $4\text{ cm}$. The finance department imposes a cost constraint: the total tinplate surface used for each closed container (including both hexagonal bases) cannot exceed $450\text{ cm}^2$.\na) Determine the area of one of the regular hexagonal bases. Remember that to calculate the apothem of the base you must use the Pythagorean theorem on the right triangle formed by the radius of the base (which measures the same as the side, $4\text{ cm}$) and half of the side. Round the apothem to two decimal places and the area of the base to the nearest tenth.\nb) From the maximum allowable total area ($450\text{ cm}^2$), set up an equation or inequality involving the height $h$ of the can and the two hexagonal bases.\nc) Solve the expression to deduce the maximum height the can can have. If the packaging machinery only allows calibrating heights in whole numbers of centimeters, what is the greatest height in centimeters that can be manufactured while remaining within the material budget?