Name: tetrahedron volume from triangle slices
Uploaded: 2026-06-30T06:21:21.659Z
Duration: 1 min 25 s
Channel: JoVE

Per i solidi in cui sono note le aree della sezione trasversale, il volume si trova integrando queste aree lungo un asse perpendicolare alle fette.Si consideri un tetraedro di altezza h e una base che è un triangolo equilatero con lato a.L'origine è posizionata al vertice superiore, con l'asse y che si estende lungo l'altezza.Le sezioni trasversali perpendicolari a questo asse formano triangoli equilateri più piccoli che aumentano di dimensione dal vertice alla base.Con un lato base parallelo all'asse x, la lunghezza del lato di ogni piccolo triangolo equilatero ad altezza y si trova usando triangoli simili. Scala come y su h per a.L'area di ogni triangolo equilatero è uguale a radice 3 su 4 volte il quadrato del suo lato. Queste aree mostrano le sezioni trasversali variabili lungo l'altezza.Il volume del tetraedro si trova integrando queste aree dal vertice alla base. Il risultato è proporzionale al prodotto tra l'altezza e il quadrato del lato della base.I geometri utilizzano questo metodo per ottenere il volume di un cumulo di ghiaia irregolare misurando sezioni trasversali variabili e integrandole per stime accurate dei materiali.

calcolo del volume tramite l’area delle sezioni trasversali

Finding Volume Using Cross-Sectional Area

Per i solidi le cui aree delle sezioni trasversali variano in modo prevedibile, il volume pu&#xF2; essere determinato integrando tali aree lungo un asse perpendicolare alle sezioni stesse. Questo approccio &#xE8; particolarmente utile per i solidi poliedrici, per i quali le formule geometriche classiche possono non essere immediatamente applicabili. Un tetraedro offre un esempio chiaro di come l&#x2019;integrazione delle sezioni trasversali possa essere applicata a un oggetto tridimensionale con geometria in continuo cambiamento.
Si consideri un tetraedro di altezza h e con base costituita da un triangolo equilatero di lato a. Il sistema di coordinate &#xE8; scelto tale che l&#x2019;origine si trovi nel vertice superiore e l&#x2019;asse y si estenda verso il basso lungo l&#x2019;altezza del tetraedro fino alla base. Le sezioni trasversali prese perpendicolarmente a tale asse formano triangoli equilateri le cui dimensioni aumentano progressivamente dal vertice alla base. Al vertice, l&#x2019;area della sezione trasversale &#xE8; nulla e raggiunge il suo valore massimo alla base.
Assumendo che un lato del triangolo di base sia parallelo all&#x2019;asse x, la lunghezza del lato di ciascun triangolo di sezione trasversale a un dato valore di y pu&#xF2; essere determinata mediante la similitudine dei triangoli. Poich&#xE9; il tetraedro si restringe in modo lineare, la lunghezza del lato di una sezione trasversale varia in proporzione diretta al rapporto tra la distanza dal vertice e l&#x2019;altezza totale. Di conseguenza, le sezioni trasversali triangolari crescono uniformemente all&#x2019;aumentare di y.
L&#x2019;area di ciascun triangolo equilatero dipende dal quadrato della lunghezza del lato, moltiplicato per un fattore geometrico costante. Tali aree descrivono come la sezione trasversale del tetraedro cambi in modo continuo lungo l&#x2019;altezza. Integrando l&#x2019;area di queste sezioni triangolari dal vertice superiore alla base, si ottiene il volume complessivo del tetraedro. Il risultato finale mostra che il volume &#xE8; proporzionale all&#x2019;altezza e al quadrato della lunghezza del lato della base, in accordo con le relazioni geometriche note.
Questa tecnica di integrazione delle sezioni trasversali trova applicazioni pratiche al di l&#xE0; dei solidi idealizzati. Topografi e ingegneri utilizzano metodi analoghi per stimare il volume di cumuli irregolari di ghiaia o di movimenti di terra. Misurando le aree delle sezioni trasversali in pi&#xF9; punti e integrandole lungo la lunghezza o l&#x2019;altezza, &#xE8; possibile determinare con accuratezza le quantit&#xE0; di materiale necessarie per la pianificazione e la realizzazione dei lavori.

Calcolo del volume tramite l’area delle sezioni trasversali

Per i solidi le cui aree delle sezioni trasversali variano in modo prevedibile, il volume pu&#xF2; essere determinato integrando tali aree lungo un asse ...

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tetrahedron volume from triangle slices

Tetrahedron volume can be found by adding the areas of triangular slices. This method uses cross-sectional area, which is the area of a flat slice taken through a 3D solid. It is useful when a solid changes shape in a regular way and a simple geometric formula is not available.

A tetrahedron is a good example. It has a height h and an equilateral triangle base with side length a. The coordinate system places the origin at the top vertex, and the y-axis goes downward along the height to the base. Slices taken perpendicular to this axis are equilateral triangles that grow larger from the vertex to the base.

At the top vertex, the cross-sectional area is zero. At the base, it is largest. Using similar triangles, the side length of a slice at height y can be found from the way the tetrahedron narrows linearly. The side length changes in direct proportion to the distance from the vertex to the total height.

Each cross-section is an equilateral triangle, so its area depends on the square of its side length and a constant geometric factor. These triangular areas show how the solid changes as it extends downward. Integrating the areas from the top vertex to the base gives the total volume of the tetrahedron. The final result matches the known relationship that volume depends on the height and the square of the base side length.

This cross-sectional integration method is also useful outside ideal math problems. Surveyors and engineers can use it to estimate the volume of irregular gravel piles or earthworks. They measure cross-sectional areas at several points and integrate along the length or height to find accurate material quantities for planning and construction.

Tetrahedron Volume from Triangle Slices

area between curves: integrating with respect to x

area-between-curves-integrating-with-respect-to-x

Vertical Slices for Area Between Curves

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup. A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point...

area between curves: integrating with respect to y

area-between-curves-integrating-with-respect-to-y

Horizontal Slices for Area Between Curves

Area between curves can be found with horizontal slices when both boundaries are written as functions of y. The left and right curves are continuous from y = c to y = d, and those lines set the vertical limits of the region. Because the curves depend on y, the area is best measured by moving across the shape with thin horizontal strips. The method uses a Riemann sum. The region is divided into many thin strips with the same height, delta y. For each strip, the width is the horizontal distance...

area between curves: problem solving

Piecewise Area Between Curves

Piecewise area between curves uses more than one integral to find the size of a region on a graph. In this example, the region is enclosed by a square root function, a reflected cube root function, and a linear function. The line intersects each of the other two curves, and those intersection points show where the boundary of the region changes. The area cannot be found with one setup over the full interval because the top and bottom curves are not the same everywhere. The region is split at...

calculation of volume of solids by integration

calculation-of-volume-of-solids-by-integration

Slicing Methods for Solid Volume

Slicing methods help find the volume of solids with changing cross-sections. A basic solid, like a rectangular box, is easy to measure because its base area stays the same along its height. Many real objects do not have that regular shape, so elementary geometry is not enough. The slicing method solves this by breaking a solid into many very thin slices. These slices are taken perpendicular to a chosen axis, often the x-axis. Each slice has a small thickness, written as delta x, and a...

volumes of solids of revolution

Disk Method for Rotated Region Volume

The disk method is used to find the volume of a solid formed by rotating a region around an axis. The region is first drawn in a two-dimensional plane. When that region spins around a fixed line, called the axis of revolution, it creates a three-dimensional shape with rotational symmetry. These shapes appear often in math models, physics, and engineering. When the region touches the axis of revolution, the solid has no hollow space inside. Its cross-sections are circular, so the shape can be...

finding volume using cross-sectional area

Tetrahedron volume can be found by adding the areas of triangular slices. This method uses cross-sectional area, which is the area of a flat slice taken through a 3D solid. It is useful when a solid changes shape in a regular way and a simple geometric formula is not available. A tetrahedron is a good example. It has a height h and an equilateral triangle base with side length a. The coordinate system places the origin at the top vertex, and the y-axis goes downward along the height to the...

problem solving: volume

Finding the Volume of a Jet Fuel Tank

The volume of a jet fuel tank can be found with solids of revolution. The tank is modeled by rotating a two-dimensional region around the x-axis. This creates a three-dimensional shape that is symmetric about the axis of rotation. The region runs from 0 to 2 meters, so the tank’s shape can be measured along that length. Because the boundary curve touches the axis, the disk method works well here. The disk method breaks the solid into many very thin circular slices that are perpendicular to the...

average value of a function

Estimating Function Behavior with Average Value

The average value of a function gives one number that summarizes how a quantity changes over a closed interval. It can also be viewed as the height of a rectangle whose area matches the net area under the curve. That net area includes both positive and negative parts of the function, so the result reflects the function’s overall behavior. A greenhouse temperature over a 24-hour period is a practical example of this idea. The temperature may rise and fall throughout the day, but it is still...

key terms and concepts

Learn how triangular cross-sections, similar triangles, and integration are used to find the volume of a tetrahedron and estimate real-world material piles.

Tetrahedron Volume from Triangle Slices | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Tetrahedron Volume from Triangle Slices

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