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

Para sólidos onde as áreas da seção transversal são conhecidas, o volume é encontrado integrando essas áreas ao longo de um eixo perpendicular às fatias.Considere um tetraedro com altura h e uma base que é um triângulo equilátero com o lado a.A origem está posicionada no vértice superior, com o eixo y se estendendo ao longo da altura.Seções transversais perpendiculares a esse eixo formam triângulos equiláteros menores que aumentam de tamanho do vértice até a base.Com um lado da base paralelo ao eixo x, o comprimento do lado de cada pequeno triângulo equilátero na altura y é encontrado usando triângulos semelhantes. Escala como y sobre h vezes a.A área de cada triângulo equilátero é igual à raiz 3 sobre 4 vezes o quadrado de seu lado. Essas áreas mostram as seções transversais variáveis ao longo da altura.O volume do tetraedro é determinado integrando essas áreas do vértice até a base. O resultado é proporcional ao produto da altura e do quadrado do lado da base.Os topógrafos utilizam esse método para obter o volume de uma pilha irregular de cascalho, medindo seções transversais variáveis e integrando-as para estimativas precisas de materiais.

cálculo do volume usando a área da seção transversal

Finding Volume Using Cross-Sectional Area

Para s&#xF3;lidos cujas &#xE1;reas de se&#xE7;&#xE3;o transversal variam de forma previs&#xED;vel, o volume pode ser determinado integrando essas &#xE1;reas ao longo de um eixo perpendicular &#xE0;s se&#xE7;&#xF5;es transversais. Essa abordagem &#xE9; particularmente &#xFA;til para s&#xF3;lidos poli&#xE9;dricos, nos quais as f&#xF3;rmulas geom&#xE9;tricas cl&#xE1;ssicas podem n&#xE3;o ser imediatamente aplic&#xE1;veis. Um tetraedro fornece um exemplo claro de como a integra&#xE7;&#xE3;o por se&#xE7;&#xF5;es transversais pode ser aplicada a um objeto tridimensional com geometria em constante mudan&#xE7;a.
Considere um tetraedro com altura h e uma base que &#xE9; um tri&#xE2;ngulo equil&#xE1;tero de lado a. O sistema de coordenadas &#xE9; escolhido de forma que a origem esteja no v&#xE9;rtice superior e o eixo y se estenda para baixo ao longo da altura do tetraedro at&#xE9; a base. Se&#xE7;&#xF5;es transversais tomadas perpendicularmente a esse eixo formam tri&#xE2;ngulos equil&#xE1;teros cujos tamanhos aumentam gradualmente do v&#xE9;rtice at&#xE9; a base. No v&#xE9;rtice, a &#xE1;rea da se&#xE7;&#xE3;o transversal &#xE9; zero e atinge seu valor m&#xE1;ximo na base.
Com um dos lados do tri&#xE2;ngulo da base paralelo ao eixo x, o comprimento do lado de cada tri&#xE2;ngulo da se&#xE7;&#xE3;o transversal, a uma dada altura y, pode ser determinado usando tri&#xE2;ngulos semelhantes. Como o tetraedro se afunila linearmente, o comprimento do lado de uma se&#xE7;&#xE3;o transversal varia em propor&#xE7;&#xE3;o direta &#xE0; raz&#xE3;o entre a dist&#xE2;ncia do v&#xE9;rtice at&#xE9; a base (na altura y) e a altura total. Consequentemente, as se&#xE7;&#xF5;es transversais em forma de tri&#xE2;ngulos crescem uniformemente &#xE0; medida que y aumenta.
A &#xE1;rea de cada tri&#xE2;ngulo equil&#xE1;tero depende do quadrado do comprimento do seu lado, multiplicado por um fator geom&#xE9;trico constante. Essas &#xE1;reas representam como a se&#xE7;&#xE3;o transversal do tetraedro varia continuamente ao longo de sua altura. Ao integrar a &#xE1;rea dessas fatias triangulares do v&#xE9;rtice superior at&#xE9; a base, obt&#xE9;m-se o volume total do tetraedro. O resultado final mostra que o volume &#xE9; proporcional &#xE0; altura e ao quadrado do comprimento do lado da base, em conson&#xE2;ncia com as rela&#xE7;&#xF5;es geom&#xE9;tricas conhecidas.
Essa t&#xE9;cnica de integra&#xE7;&#xE3;o de se&#xE7;&#xF5;es transversais tem aplica&#xE7;&#xF5;es pr&#xE1;ticas al&#xE9;m de s&#xF3;lidos idealizados. Top&#xF3;grafos e engenheiros utilizam m&#xE9;todos semelhantes para estimar o volume de pilhas irregulares de cascalho ou obras de terraplenagem. Ao medir as &#xE1;reas das se&#xE7;&#xF5;es transversais em diversos pontos e integrar ao longo do comprimento ou da altura, podem ser determinadas quantidades precisas de material para fins de planejamento e constru&#xE7;&#xE3;o.

Cálculo do volume usando a área da seção transversal

Para s&#xF3;lidos cujas &#xE1;reas de se&#xE7;&#xE3;o transversal variam de forma previs&#xED;vel, o volume pode ser determinado integrando essas ...

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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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