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Sum of consecutive (triangular) cubes (Nicomachus's theorem)

In number theory, the sum of the first n cubes is the square of the nth triangular number. The sequence of squared triangular numbers is

0, 1, 9, ... more

Volume of a Square Pyramid

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.

... more

Vickers hardness number

The Vickers hardness test was developed in 1921 by Robert L. Smith and George E. Sandland at Vickers Ltd as an alternative to the Brinell method to measure ... more

Volume of a pyramid (The base is a regular polygon)

A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid ... more

Pyramid frustum's volume (n-sided regular polygon bases)

A pyramid frustum is the portion of a pyramid that lies between two parallel planes cutting it. The volume of a pyramid frustum, whose bases are n-sided ... more

Volume of a pyramid

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a ... more

Pyramid frustum (volume)

A pyramid frustum is the portion of a pyramid that lies between two parallel planes cutting it.
The volume of a pyramid frustum is related with the ... more

Worksheet 341

The awe‐inspiring Great Pyramid of Cheops was built more than 4500 years ago. Its square base, originally 230 m on a side, covered 13.1 acres, and it was 146 m high (H), with a mass of about 7×10^9 kg. (The pyramid’s dimensions are slightly different today due to quarrying and some sagging). Historians estimate that 20,000 workers spent 20 years to construct it, working 12-hour days, 330 days per year.

a) Calculate the gravitational potential energy stored in the pyramid, given its center of mass is at one-fourth its height.

Division
Potential energy

b) Only a fraction of the workers lifted blocks; most were involved in support services such as building ramps, bringing food and water, and hauling blocks to the site. Calculate the efficiency of the workers who did the lifting, assuming there were 1000 of them and they consumed food energy at the rate of 300 Kcal/hour.

first we calculate the number of hours worked per year.

Multiplication

then we calculate the number of hours worked in the 20 years.

Multiplication

Then we calculate the energy consumed in 20 years knowing the energy consumed per hour and the total hours worked in 20 years.

Multiplication
Multiplication

The efficiency is the resulting potential energy divided by the consumed energy.

Division
Slant height of a right regular pyramid

Slant height is the height of any slant triangle of the regular pyramid, or the distance from the apex, down the side, to a point on the base ... more

Sum of natural numbers raised to the fourth power

The formula is a special case of general Faulhaber’s formula and gives the sum of natural consecutive numbers raised to the fourth power,(starting ... more

Volume of a tetrahedron

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. It has six edges and four ... more

Sum of natural numbers raised to the fifth power

he formula is a special case of general Faulhaber’s formula and gives the sum of natural consecutive numbers raised to the fifth power,(starting with 1), ... more

Volume of a bipyramid (base regular n-sided polygon)

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal
pyramid and its mirror image base-to-base.
The volume of a ... more

Height of a regular tetrahedron

The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. A ... more

Magic square

In recreational mathematics, a magic square is an arrangement of distinct numbers, usually integers, in a square grid, where the numbers in each row, and ... more

N-pointed Magic Star

An n-pointed magic star is a star polygon in which numbers are placed at each of the n vertices and n intersections, such that the four numbers on each ... more

Surface area of a regular tetrahedron

The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. A ... more

Magic cube (simple)

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the ... more

Surface area of a right regular pyramid

The surface of a right regular pyramid can be calculated by the area and the perimeter of the base and the slant height

... more

Knoop hardness test

The Knoop hardness test /kəˈnuːp/ is a microhardness test – a test for mechanical hardness used particularly for very brittle materials or thin sheets, ... more

Optimum pyramidal horn antenna - H-field

A horn antenna or microwave horn is an antenna that consists of a flaring metal waveguide shaped like a horn to direct radio waves in a beam. Horns are ... more

Optimum pyramidal horn antenna - E-field

A horn antenna or microwave horn is an antenna that consists of a flaring metal waveguide shaped like a horn to direct radio waves in a beam. Horns are ... more

Total Area of a Frustum of a Right Circular Cone

In geometry, a frustum is the portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it.The total area of a frustum ... more

Magic hypercube

In mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts; that is, a number of integers ... more

Variance (regarding to the arithmetic mean)

The variance measures how far a set of numbers of n equally likely values is spread out. A small variance indicates that the data tend to be very close to ... more

Arithmetic progression

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is ... more

Square wave (frequency)

A square wave is a non-sinusoidal periodic waveform (which can be represented as an infinite summation of sinusoidal waves), in which the amplitude ... more

Euler's quadrilateral theorem

In any convex quadrilateral the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of ... more

Standard deviation of any arithmetic progression

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant and is ... more

Square Root

calculates the square root of a number

... more

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