Welcome to our page dedicated to area and volume problems! Here you will find a collection of challenging mathematical problems related to the measurement of areas and volumes.

First of all, we must know that area is a measure that describes the extension of a two-dimensional surface. It represents the amount of space occupied by a figure in a plane and is expressed in square units, such as square feet (ft²) or square inches (in²).

On the other hand, volume is a measure that describes the amount of three-dimensional space occupied by an object. It represents the space enclosed within a figure or a solid and is expressed in cubic units, such as cubic feet (ft³) or cubic inches (in³). Of course, area and volume vary according to the object.

These quantities are fundamental in many disciplines, from architecture and engineering to physics and geometry. Here, we will provide you with clear and step-by-step explanations of how to calculate the area and volume of different shapes and solids that appear in mathematical and/or everyday life problems.

Let's go! Get ready to develop your problem-solving skills. Enjoy and learn from the techniques we have developed for you.

Score:

Calculate the volume, in cubic centimeters, of a room that is $\text{[math]}$ m (16.4 ft) long, $\text{[math]}$ m (13.1 ft) wide and $\text{[math]}$ m (8.2 ft) high

Solution

Representacion gráfica de una habitacion de 5 por 4 por 2.4 metros

1 We calculate the volume

$\text{[math]}$

Knowing that $\text{[math]}$ , we convert:

$\text{[math]}$

A pool is $\text{[math]}$ m (26.2 ft) long, $\text{[math]}$ m (19.7 ft) wide and $\text{[math]}$ m (4.9 ft) deep. The pool is painted at a rate of $ $\text{[math]}$ per square meter

A How much will it cost to paint it.
B How many liters of water will be needed to fill it.

Solution

Representacion gráfica de una piscina de 8 por 6 por 1.5 metros

1 We calculate the area to paint

$\text{[math]}$

2 We calculate the cost

$\text{[math]}$

3 The liters needed to fill it is the volume of the pool multiplied by 1000

$\text{[math]}$

In a warehouse with dimensions $\text{[math]}$ m (16.4 ft) long, $\text{[math]}$ m (9.8 ft) wide and $\text{[math]}$ m (6.6 ft) high, we want to store boxes with dimensions $\text{[math]}$ dm (3.3 ft) long, $\text{[math]}$ dm (2.0 ft) wide and $\text{[math]}$ dm (1.3 ft) high. How many boxes can we store?

Solution

grafica de un almacen de 5 por 3 por 2 metros

1 First we observe that

$\text{[math]}$

2 We calculate the volume of the warehouse

$\text{[math]}$

3 We calculate the volume of the boxes

$\text{[math]}$

4 The number of boxes is obtained by dividing the warehouse volume by the volume of one box

$\text{[math]}$ boxes

Determine the total area of a tetrahedron, an octahedron and an icosahedron with $\text{[math]}$ cm (2.0 in) edge

Solution

1 The total area of a tetrahedron is

$\text{[math]}$

2 The total area of an octahedron is

$\text{[math]}$

3 The total area of an icosahedron is

$\text{[math]}$

Calculate the height of a prism that has a base area of $\text{[math]}$ dm² (1.29 ft²) and $\text{[math]}$ l (12.7 gal) capacity

Solution

1 We have that $\text{[math]}$ is equivalent to $\text{[math]}$ of volume

2 We calculate the volume of the prism

$\text{[math]}$

3 We equate both volumes

$\text{[math]}$

Calculate the amount of tin that will be needed to make $\text{[math]}$ cylindrical cans with $\text{[math]}$ cm (3.9 in) diameter and $\text{[math]}$ cm (7.9 in) height

Solution

Ejercicios de areas y volumenes grafica de un bote de 20 centimetros de altura

1 We calculate the total area of one can

$\text{[math]}$

2 The amount used for 10 cans is

$\text{[math]}$

A cylinder has a height equal to the circumference of the base. And the height measures $\text{[math]}$ cm (49.5 in). Calculate:

A Total area.

B Volume.

Solution

1 We calculate the radius

$\text{[math]}$

2 We calculate the total area

$\text{[math]}$

3 We calculate the volume

$\text{[math]}$

In a test tube with $\text{[math]}$ cm (2.4 in) radius, four ice cubes with $\text{[math]}$ cm (1.6 in) edge are placed. What height will the water reach when they melt?

Solution

1 We calculate the volume of the ice cubes

$\text{[math]}$

2 The test tube is cylindrical, so its volume is

$\text{[math]}$

3 We equate the volumes and obtain

$\text{[math]}$

The dome of a cathedral has a hemispherical shape, with radius $\text{[math]}$ m (164 ft). If restoring it costs $ $\text{[math]}$ per m², how much will the restoration budget amount to?

Solution

1 We calculate the area of the hemisphere

$\text{[math]}$

2 The restoration cost is

$\text{[math]}$

How many square tiles with $\text{[math]}$ cm (7.9 in) sides are needed to cover the faces of a pool $\text{[math]}$ m (32.8 ft) long by $\text{[math]}$ m (19.7 ft) wide and $\text{[math]}$ m (9.8 ft) deep?

Solution

Ejercicios de areas y volumenes dibujo de piscina que mide 10 por 6 por 3 metros

1 We calculate the area to cover

$\text{[math]}$

2 We calculate the area of one tile

$\text{[math]}$

3 The number of tiles required is

$\text{[math]}$

A cylindrical container with $\text{[math]}$ cm (3.9 in) radius and $\text{[math]}$ cm (2.0 in) height is filled with water. If the mass of the full container is $\text{[math]}$ kg (4.4 lb), what is the mass of the empty container?

Solution

1 We calculate the volume

$\text{[math]}$

2 The weight of the container is

$\text{[math]}$

For a party, Louis has made $\text{[math]}$ cone-shaped hats with cardboard. How much cardboard will he have used if the hat dimensions are $\text{[math]}$ cm (5.9 in) radius and $\text{[math]}$ cm (9.8 in) slant height?

Solution

Ejercicios de areas y volumenes dibujo de cono

1 We calculate the area of one cone

$\text{[math]}$

2 The area required for 10 cones is

$\text{[math]}$

A cube with $\text{[math]}$ cm (7.9 in) edge is full of water. Would this water fit in a sphere with $\text{[math]}$ cm (7.9 in) radius?

Solution

1 We calculate the volume of the cube

$\text{[math]}$

2 We calculate the volume of the sphere

$\text{[math]}$

Since the volume of the sphere is greater than the volume of the cube, we conclude that the water does fit in the sphere.

Calculate the diagonal of a rectangular prism $\text{[math]}$ cm (3.9 in) long, $\text{[math]}$ cm (1.6 in) wide and $\text{[math]}$ cm (2.0 in) high

Solution

Ejercicios de areas y volumenes grafica de una caja de 10 por 5 por 4 centimetros

1 The diagonal is given by

$\text{[math]}$

A sphere and a cube are filled with water at the same time at a rate of 1m³ per minute. If the sphere has a radius of $\text{[math]}$ meters (16.4 ft) and the cube edge measures $\text{[math]}$ meters (26.2 ft), which will fill first?

Solution

To solve this problem we must calculate the volume of each container. The formula for the volume of a sphere is

$\text{[math]}$

If $\text{[math]}$ , then the volume of the sphere is

$\text{[math]}$

Similarly, the volume of a cube with edge $\text{[math]}$ is given by

$\text{[math]}$

Thus, if $\text{[math]}$ , then the volume of the cube is

$\text{[math]}$

Since both containers are filled at the same rate of 1m³ per minute, then the sphere will fill in approximately 524 minutes or in 8 hours and 44 minutes. Similarly, the cube will fill in 512 minutes or in 8 hours and 32 minutes. Therefore the cube will fill with water first.

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Agostina

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.

Area and Volume Problems

Area and Volume Exercises and Problems

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Area and Volume Problems

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Area and Volume Exercises and Problems

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