Volume of Pyramid

Do you know that the Great Pyramid of Giza measures about 146.7 m high and 230.6 m in base length? Can you imagine how many cubes of sugar measuring 1 m3 would be needed to fill the Great Pyramid of Giza? Herein, you shall be learning about how this can be calculated through the knowledge of the volume of pyramids.

Volume of Pyramid Volume of Pyramid

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Contents
Table of contents

    What is a pyramid?

    Pyramids are 3-dimensional objects with triangular sides or surfaces that meet at a tip called an apex. The name 'pyramid' often brings to mind the Pyramids of Egypt, which is one of the seven wonders of the world.

    In geometry, a pyramid is a polyhedron obtained connecting a polygonal base to a point, called the apex.

    Types of pyramids

    Pyramids are of various types depending on the shape of their base. A pyramid with a triangular base is called a triangular pyramid, and a rectangular-based pyramid is known as a rectangular pyramid. The sides of a pyramid are triangular and they emerge from its base. They all meet at a point called the apex.

    Volume of Pyramids An image showing the various types of pyramids StudySmarterAn image showing the various types of pyramids, Njoku - StudySmarter Originals

    What is the volume of a pyramid?

    You may be wondering how many blocks of sand can make up the Egyptian pyramids. The volume of a pyramid is the space enclosed by its faces. Generally, the volume of a pyramid is a-third of its corresponding prism. Its corresponding prism has the same base shape, base dimensions and height. Thus, the general formula for calculating the volume of a pyramid is,

    V=13×bh

    where,

    V is the volume of the pyramid

    b is the base area of pyramid

    h is the height of pyramid

    Note that this is the general formula for the volume of all pyramids. Differences in the formulas are based on the shape of the base of the pyramid.

    Volume of rectangular pyramids

    The volume of rectangular pyramids can be found by multiplying a third of the rectangular base area by the height of the pyramid. Therefore:

    Volume of rectangular pyramid=13×base Area×heightBase area=length×breadthVolume=13×l×b×h

    where;

    l is the length of the base

    b is the breadth of the base

    h is the height of the pyramid

    Volume of Pyramids An illustration of the sides of a rectangular pyramid StudySmarterAn illustration of the sides of a rectangular pyramid, Njoku - StudySmarter Originals

    This means that the volume of a rectangular pyramid is a third of the corresponding rectangular prism.

    Volume of square-base pyramids

    A square base pyramid is a pyramid whose base is a square. The volume of square-based pyramids can be gotten by multiplying one-third of the square base area by the height of the pyramid. Therefore:

    Volume of square base pyramid=13×base Area×heightBase area=length2Volume=13×l2×h

    where;

    l is the length of the square base

    h is the height of the pyramid

    Volume of Pyramids An illustration of the sides of a square base pyramid StudySmarterAn illustration of the sides of a square base pyramid, Njoku - StudySmarter Originals

    Volume of triangular-based pyramids

    The volume of triangular base pyramids can be obtained by multiplying one-third of the triangular base area by the height of the pyramid. Therefore:

    Volume of triangular base pyramid=13×base Area×heightBase area=12×base length×height of triangleVolume=13×12×b×htriangle×hpyramidV=16×b×htriangle×hpyramid

    where;

    l is the length of the base

    b is the triangular base length

    htriangle is the height of the triangular base

    hpyramid is the height of the pyramid

    Volume of Pyramids An illustration of the sides of a triangular pyramid StudySmarterAn illustration of the sides of a triangular pyramid, Njoku - StudySmarter Originals

    Volume of hexagonal pyramids

    The volume of hexagonal base pyramids can be gotten by multiplying one-third of the hexagonal base area by the height of the pyramid. Therefore:

    Volume of triangular base pyramid=13×base Area×heightBase area=332×length2Volume=13×332×l2×hVolume=32×l2×h

    Volume of Pyramids An illustration of the sides of a hexagonal pyramid StudySmarter An illustration of the sides of a hexagonal pyramid, Njoku - StudySmarter Originals

    A pyramid of height 15ft has a square base of 12 ft. Determine the volume of the pyramid.

    Solution

    Volume of square base pyramid=13×l2×hl=12fth=15ftV=13×122×15V=5×144V=720ft3

    Calculate the volume of the figure below:

    Composition of Pyramid and Cuboid

    Solution

    The volume of the figure=volume of rectangular pyramid + volume of rectangular prismVolume of rectanglar pyramid= 13×l×b×hl=45 cmb=20 cmh=50 cmVolume of rectanglar pyramid= 13×45×20×50Volume of rectanglar pyramid= 15000 cm3Volume of rectangular prism=l×b×hl=45 cmb=20 cmh=40 cmVolume of rectangular prism=45×20×40Volume of rectangular prism=36000 cm3The volume of the figure=volume of rectangular pyramid + volume of rectangular prismThe volume of the figure=15000+36000The volume of the figure=51000 cm3

    A hexagonal pyramid and a triangular pyramid are of the same capacity. If its triangular base has a length of 6 cm and a height of 10 cm, calculate the length of each side of the hexagon when both pyramids have the same height.

    Solution

    The first step is to express the relationship in an equation.

    According to the problem, the volume of the triangular pyramid equals the volume of the hexagonal pyramid.

    Let bt signify the base area of triangular base and bh represent the base area of hexagonal base.

    Then:

    Volume of triangular pyramid=Volume of hexagonal pyramidbth3=bhh3

    Multiply both sides of the equation by 3 and divide by h.

    bth3=bhh3bth3×3h=bhh3×3hbt=bh

    This means that the triangular base and the hexagonal base are of equal area.

    Recall that we are required to find the length of each side of the hexagon.

    bt=12×base length×heightbase length of triangle=6 cmheight of triangle=10 cmbh=332×l2

    Where l is the length of the side of a hexagon.

    Recall that bt = bh, then;

    12×6×10=332×l212×6×10×233=332×l2×233203=l2

    Take the roots of both sides of the equation.

    l2=11.547l=3.398 cm

    Thus each side of the hexagonal base is approximately 3.4 cm.

    Volume of Pyramid - Key takeaways

    • A pyramid is a 3-dimensional object with triangular sides or surfaces that meet at a tip called an apex
    • The various types of pyramids are based on the shape of their base
    • The volume of a pyramid is one-third the base area × height
    Frequently Asked Questions about Volume of Pyramid

    What is the volume of a pyramid? 

    It is the capacity of a pyramid or the space it contains.

    What formula is used to determine the volume of a pyramid?

    The formula used in calculating the volume of a pyramid is one-third the volume of the corresponding prism.

    How do you calculate the volume of a pyramid with a square base? 

    The volume of a pyramid with a square base is calculated by finding the product of one-third of the area of one of the square bases and the height of the pyramid.

    How do you calculate the volume of a pyramid with a triangular base? 

    The volume of a pyramid with a triangular base is gotten by multiplying one third of the triangular base area by the height of the pyramid.

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