Ken Ward's Mathematics Pages

Four-sided

Four sided figures are called quadrilaterals.

A quadrilateral with opposite sides equal and parallel is called a parallelogram. The corresponding solid is called a parallelepiped. A parallelogram, with all its angles right angles is called a rectangle. The solid version of the rectangle is called a cuboid. A rectangle with all its sides equal is called a square, which is, in 3d, a cube.

The triangle creeps into this page because we need to know its area in order to compute the area of a trapezoid.

A quadrilateral with only two of its sides parallel is called a trapezium, or trapezoid. There is a confusing difference in meaning between the USA and elsewhere with these terms, where a trapezium in the USA means a quadrilateral with no sides parallel.

Proclus, who wrote the commentaries for Euclid's Elements, called a trapezium  figure with exactly two sides parallel, and a trapezoid one with no sides parallel. However, in 1795, Hutton's Mathematical Dictionary accidentally reversed the two definitions, and in USA and other countries, the trapezium became a trapezoid and the trapezoid became a trapezium.

Square

A square is a figure with all its sides equal and all its angles right-angles.

The area of a square, side a, is a², and its periphery is 4a. The square is fundamental to area, and areas are measures of the number of unit squares. We find the area of a square by counting the number of unit squares in the square.

Rectangle

A figure with opposite sides equal and parallel, and all its angles right-angles is called a rectangle.

The area of the rectangle is ab, and its periphery is 2(a+b).
Naturally, a square is a special case of a rectangle.

We find the area of a rectangle by counting the number of unit squares. For instance:
The rectangle below is made up of 3 rows each containing 4 unit squares. The number of unit squares is therefore 3x4, or 12. This is a rectangle of 12 unit squares, or a rectangle covering an area of 12 unit squares. By generalising this, for the above rectangle, it is a rectangle made of b rows of a unit squares. So it has ab unit squares. Or its area is ab.

We have allowed a and b to be any real number.

Cuboid

A rectangle in 3 dimensions is called a cuboid.

The volume of a cuboid is abc.
Its surface area is 2ab+2bc+2ac

A cube is a special case of a cuboid. It has all its sides equal.
The volume of a cube is a³.
Its surface area is 6a²

In the cuboid below, each of the "blocks" are unit cubes. The volume in therefore the sum of the unit cubes.

If we count all the blocks, we get the volume of the cuboid. We may think of the cuboid as three sets of the "face" of the cuboid, which is an area of 4x3 or 12. The volume is therefore 12 blocks.

Alternatively, we can think of the cuboid as being 3 rows of the base, which is 4x3 cubes. Or four columns of either end, which is 3x3 cubes.

Parallelogram

A parallelogram is a four-sided figure with its opposite sides parallel and equal.

The figure ABCD is a parallelogram. The height of the parallelogram is h.
The area of the parallelogram is ah.
The periphery is 2(a+b).

ABCD is a parallelogram, shown in blue. DX is vertical to AB and BY is vertical to DC. So, BCY and XDC are right angles.
The end triangle BCY is equal to the triangle ADZ, so we can chop off BCY and place it in the position ADZ. We then have a rectangle, ABYZ, equal in area to the parallelogram ABCD. So the area of ABCE equals the area of ABYZ , which is ABxXD, or using the value from the previous diagram, the area is ah, where a=AB, and h=XD.

The triangles BCY and ADZ are equal because they are congruent in the Euclidean sense (SAS, AA corresponding side).
Angle BCY=Angle ADZ (corresponding angles, ZC parallel to AB). The right angles are equal, so, the angles YBC and ZAD are equal (angle sum of a triangle).  ZA=YB (perpendicular distance between two parallel lines), and AC and BC are equal (two parallel lines cut by two parallel lines). Therefore, the two triangles are equal, having one angle equal between two equal sides (SAS).

Parallelepiped

When we extend a rectangle into space, we get a cuboid; when we do the same thing with a parallelogram we get a parallelepiped.
The volume of a parallelepiped is Ah, where A is the area of the base, and h is the vertical height of the parallelepiped.

As their base is a rectangle, its area is ab, so:
The volume of the parallelepiped is abh.

Triangle

The area of a triangle with a base, b, and a height, h, is bh/2.
In the diagram below, ABC is a triangle. BY is parallel to AC. XY is parallel to AZ (which is AB extended). Therefore, ABYC is a parallelogram. The area of the parallelogram is AB.h, where h is the vertical height. h is also the vertical height of the triangle ABC.


Parallel lines are marked with arrows, and equal angles with the same number of squiggly lines. If in the parallelogram, the triangle ABC is equal to the triangle BYC, then because we know the area of the parallelogram is AB.h, then the area of each triangle is AB.h/2.

If angle BAC=angle BYC,
angle ACB=CBY, then
because CB=CB (common side of the triangles and corresponding to the angles)
the triangles ABC and BYC are congruent, or equal in all respects.


Angle BYC=angle ACX (corresponding angles on the parallels XY and AZ).
Angle ACX==angle BAC (alternate angles on the parallels XY and AZ)
So, angle BAC=angle BYC.

Angle ACB=angle CBY (alternate angles on  the parallels AC and BY)

Because two of the angles are equal in the two triangles and the corresponding side is equal, then the two triangles are congruent and so their area is half that of the parallelogram ABYC, and is therefore AB.h/2, or half the base times the height

Trapezium, Trapezoid

The trapezium is a four-sided figure with only two of its sides parallel:

The area, A, of the trapezium is 
Which is the average length of the parallel sides (AB and CD) times the distance between them, or the height.
The diagram above is a trapezium.
The area is:
Area rectangle DXYC+area triangle AXD+area triangle YBC
Or:

Noting that XY=b, and AX+YB=a-b

And simplified becomes:









Ken Ward's Mathematics Pages

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Faster Arithmetic - by Ken Ward