Thứ Ba, Tháng Hai 11, 2025
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HomeTechnologyFormula for calculating the volume of a rectangular prism

Formula for calculating the volume of a rectangular prism

A prism is a polygon with two parallel and equal base faces and a parallelogram on the sides.

Regular quadrilateral prism

Comment:

  • The sides of the prism are equal and parallel to each other
  • The sides are parallelograms
  • The two bases of a prism are two equal polygons

Formula to calculate the volume of a prism (V prism), what is the formula to calculate the volume of a vertical prism? Please refer to the article below.

1. Volume of a vertical prism

Formula for calculating the volume of a vertical prism:

The volume of a vertical prism is equal to the product of the area of ​​the base multiplied by the height.

V = Bh

In there

  • V is the volume of the prism (unit m3)
  • B is the base area (unit m2)
  • h is the height of the prism (unit m)

3. Classification of prisms

Regular prismatic shape

It is a vertical prism with a base that is a regular polygon. The side faces of the prism are all equal rectangles. For example: equilateral triangular prisms, equilateral quadrilaterals… then we understand them as equilateral prisms

Equilateral triangular prism

A base surface that is a regular quadrilateral is called a regular quadrilateral prism.

Regular quadrilateral prism

Triangular prism shape

  • A triangular prism has 5 faces, 9 edges, and 6 vertices.
  • The two bases are triangular and parallel to each other; Each side is rectangular;
  • The sides are equal;
  • The height of a triangular prism is the side length.

For example:

Triangular prism shape

The triangular prism ABC.A'B'C' has:

– The lower base is triangle ABC, the upper base is triangle A'B'C';

The sides are rectangles: AA'B'B, BB'C'C, CC'A'A;

– Edges:

  • Bottom edge: AB, BC, CA, A'B', B'C', C'A'
  • Side: AA', BB', CC';
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– Vertices: A, B, C, A', B', C'.

– Height is the length of one side: AA' or BB' or CC'.

Quadrilateral vertical prism

– A quadrilateral prism has 6 faces, 12 edges, and 8 vertices.

– The two base faces are quadrilaterals and parallel to each other. Each side is rectangular.

– The sides are equal.

– The height of a quadrilateral prism is the length of one side.

For example:

Quadrilateral vertical prism

Quadrilateral prism ABCD.A'B'C'D' has:

– The lower base is quadrilateral ABCD, the upper base is quadrilateral A'B'C'D';

The sides are rectangles: AA'B'B, BB'C'C, CC'D'D, DD'A'A;

– Edges:

+ Bottom edge: AB, BC, CD, DA, A'B', B'C', C'D', D'A'

+ The sides: AA', BB', CC', DD' are equal.

– Vertices: A, B, C, D, A', B', C', D'.

– Height is the length of one side: AA' or BB' or CC' or DD'.

Note: Rectangles and cubes are also quadrilateral prisms.

Rectangles and cubes are also quadrilateral prisms.

Vertical prism

If a prism has side edges perpendicular to the base surface, it is called a vertical prism.

Vertical prism

Note:

If the base surface is rectangular, the vertical cylinder of the quadrilateral has another name, a rectangular box.

If a quadrilateral cylinder has 12 sides of length a, then its name is a cube.

Compare a vertical prism and a regular prism:

DEFINE: NATURE
+ A vertical prism is a prism with one side perpendicular to the base

+ The sides of a vertical prism are rectangular

+ The prismatic side faces stand perpendicular to the bottom surface

+ Height is the side

+ A regular prism is a vertical prism whose base is a regular polygon

+ The side faces of the prism are all equal rectangles

+ Height is the side

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4. Example of calculating the volume of a vertical prism

Example 1:

Let the prism ABC.A'B'C' have base ABC as an equilateral triangle with side a = 2 cm and height h = 3 cm. Calculate the volume of this prism?

The prism ABC.A'B'C' has base ABC as an equilateral triangle

Prize:

Because the base is an equilateral triangle with side a, the area is: S_{ABC}=a^2 \cdot \frac{\sqrt{3}}{4}=2^2 \cdot \frac{\sqrt{3}}{4}=\sqrt{3}\left(m ^2\right)

At this point, the volume of the prism is:

V=S_{ABC} \cdot h=\sqrt{3} \cdot 3=3 \sqrt{3}\left(m^3\right)

Example 2:

Exercise 1: Given a vertical box with sides AB = 3a, AD = 2a, AA'= 2a. Calculate the volume of block A'.ACD'

Instruct:

  Let the box be vertical

Because the side ADD'A' is rectangular, we have:

S_{AA^{\prime} D^{\prime}}=\frac{1}{2} S_{AA^{\prime} D^{\prime} D}

V_{A^{\prime} \cdot ACD^{\prime}}=V_{C \cdot AA^{\prime} D^{\prime}}=\frac{1}{2} V_{C \cdot AA^{\prime} D^{\prime} D}

=\frac{1}{2} \cdot \frac{1}{3} V_{ABCD \cdot A^{\prime} B^{\prime} C^{\prime} D^{\prime}}

=\frac{1}{6} \cdot 3 a \cdot 2 a \cdot 2 a=2 a^3

Example 3: Let the right prism ABC.A'B'C' have the base as an equilateral triangle of side a√3, the middle and base angles are 60º. Call M the midpoint of BB'. Calculate the volume of the pyramid M.A'B'C'.

Prize:

Rectangular prism ABC.A'B'C'

Due AA^{\prime} \perp(ABC) should infer

\left(\mathrm{A}^{\prime} \mathrm{C},(\mathrm{ABC})\right)=\widehat{A^{\prime} CA}=60^{\circ}

We have: AA^{\prime}=AC \cdot \tan \widehat{A^{\prime} CA}=a \sqrt{3} \cdot \tan 60^{\circ}=3 a

S_{A^{\prime B}{ }^{\prime \prime} C^{\prime}}=\frac{(a \sqrt{3})^2 \sqrt{3}}{4}=\ frac{3 a^2 \sqrt{3}}{4}

MB^{\prime}=\frac{AA^{\prime}}{2}=\frac{3 a}{2}

\Rightarrow V_{M \cdot A^{\prime} B^{\prime} C^{\prime}}=\frac{1}{3} MB^{\prime} \cdot S_{A^{\prime } B^{\prime} C^{\prime}}=\frac{3 a^2 \sqrt{3}}{8}

Example 4:

Given a regular quadrilateral prism ABCD.A'B'C'D' with base edge equal to a and face (DBC') with base ABCD at an angle of 60º. Calculate the volume of prism ABCD.A'B'C'D?

Regular quadrilateral prism ABCD.A'B'C'D'

We have: AC ⊥ BD at center O of square ABCD.

On the other hand CC' ⊥ BD therefore BD ⊥ (COC')

Inferred ((C'BD),(ABCD)) = ∠(C'OD) = 60º

Again there is:

OC=\frac{AC}{2}=\frac{a \sqrt{2}}{2}

\Rightarrow CC^{\prime}=OC \cdot \tan \widehat{C^{\prime} OD}=\frac{a \sqrt{2}}{2} \cdot \tan 60^{\circ}=\frac{a \sqrt{6}}{2}

V_{ABCD \cdot A^{\prime} B^{\prime} C^{\prime} D^{\prime}}=S_{ABCD} \cdot CC^{\prime}

=a^2 \cdot \frac{a \sqrt{6}}{2}=\frac{a^3 \sqrt{6}}{2}

Example 5:

Calculate the volume V of cube ABCD.A'B'C'D', knowing AC'=a√3

Cube ABCD.A'B'C'D'

Prize:

Let x be the side length of the cube

Consider triangle AA'C right-angled at A with:

Therefore, the volume of the cube is V=a^3.

In addition to the formula for calculating the volume of a prism above, you can refer to the article about the formula for calculating the volume of a circular block, the formula for calculating the area and circumference of a circle…

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