Geometry for Elementary School/Plane shapes
In this section, we will talk about plane figures, which are formed with coplanar (on the same plane) points joined together. When planes run into each other, the intersect. The line produced in between is called the line of intersection.
Any shape that can be drawn in the plane is called a plane figure. A shape with only straight sides as edges is called a polygon(POL-ee-gone). Polygons must have at least three sides, thus the polygons with the fewest number of sides are triangles. Circles and semicircles are not polygons because they have curved sides.
When all the sides of a polygon are equal, it is equilateral (ee-quee-LAH-teh-roll). When all the angles of a polygon are equal, it is equiangular (ee-quee-ANG-ger-lah). When a polygon is both equilateral and equiangular, it is a regular shape. When doing mathematics problems, it is very important that an equilateral shape may not be equiangular (such as a rhombus), and an equiangular shape may not be equilateral (such as a rectangle). However, an equilateral triangle is always both (see below).
When dealing with plane figures, there are two measurements that are important to find: the area and the perimeter. The perimeter is the length around the shape while the area is the size of the shape. They can be calculated with different formulae.
|The corresponding material in Euclid's elements can be found on page 27 of Book I, Definition(s) 24-29 in Issac Todhunter's 1872 translation, The Elements of Euclid for the Use of Schools and Colleges.|
A triangle is a shape with three sides. It can be classified according to its sides or angles, with three kinds each. Here they are:
- Equilateral triangles, which are also equiangular triangles, have three sides equal and three angles equal. Their angles are always 60°.
- Isosceles triangles are triangles in which two of the sides are equal. The non-included angles of the sides are also equal.
- Scalene triangles have no equivalence in any way.
- Right triangles are triangles with a right angle. The longest side of such triangles is called a hypotenuse.
- Obtuse triangles are triangles with an obtuse angle.
- Acute triangles are triangles with no right or obtuse angle.
It is interesting to note that the interior angles of triangles must add up to 180°. This is commonly used in proofs and other problems. Imagine a triangle whose points are marked A, B and C, angle A is 60 degrees, and angle B is 70 degrees:
Usually, when drawing a triangle, we draw one side horizontally. This side is usually called the base. There is nothing special about the base. By turning your paper you can make any side into the base. There is no mathematical reason to call one side a base; we do it to make talking about the triangle easier. When you have a triangle and think of one of the sides as the base, then there is one corner of the triangle that is not on the base and this point is the furthest point on the triangle from the base. The height of the triangle is the line that is perpendicular to the base and goes through that furthest point. Sometimes instead of being called the height it is called the altitude of the triangle. (So if your teacher calls it an altitude, don't worry, it's really the same thing.) The length of the base and the height are the only two numbers you need to know when calculating the area of any triangle. Just multiply base and height and divide by two (or multiply it by a half if you like.) and you have the area of the triangle!
The perimeter of the triangle is easy: just add up all the sides and voilà, you have the perimeter. You can multiply one side of an equilateral triangle by three as well. As for isosceles triangles, simply multiply one of the equal sides by two and add the shorter one. There we go.
A quadrilateral is a shape with four sides. You will spend a lot of time with these. They can be classified into many different categories:
- Parallelograms are shapes where opposite sides and angles are equal. The opposite sides are parallel, hence the name.
- Rectangles are parallelograms where the angles are all 90°. Its width or breadth refers to the shorter sides, while its length refers to its longer ones.
- Rhombuses are parallelograms where all the sides are equal, and opposite angles are equal.
- Squares are parallelograms that are both rectangles and rhombuses, i.e. all angles are right and all sides are equal.
- Trapeziums, called trapezoids in American English, have two opposite sides that are parallel. The parallel sides are sometimes called the upper and lower bases.
- Right-angles trapeziums are trapeziums with a right angle.
- Isosceles trapeziums are trapeziums where the laterals sides are equal but not parallel.
- Scalene trapeziums are trapeziums that fall into neither category.
- Kites are quadrilaterals where two pairs of adjacent sides are equal and one pair of opposite angles is equal.
- Irregular quadrilaterals are any quadrilaterals that do not fit into one of the groups above.
Calculating the area of these shapes can be very easy. For parallelograms, simply multiply the base with the height, the way with do with triangles, except we don't need to divide by two. The square is especially easy: just square one of the sides, which would be the length. For the others, we can cut them up into bite-sized pieces before we calculate. For example, we can dissect the right-angled trapeziums into a right-angled triangle and a rectangle.
The perimeter of these shapes are just as easy. For rectangles, we simply add up the length and the width, then multiply by two. You can simply multiply the length of a square by four. The isosceles trapeziums are just as easy: multiply one of the lateral sides by two, then add it up with the other two. The kite is easy as well: Just add up the two different sides and multiply that by two. For the rest, you can just add up everything.
Many other polygons have a name. The following are the ones you need to know in elementary school:
- Pentagons have five sides.
- Hexagons have six sides.
- Heptagons or septagons have seven sides.
- Octagons have eight sides.
- Nonagons have nine sides.
- Decagons have ten sides.
And here are two more extras:
- Hendecagons (also known as undecagons) have eleven sides.
- Dodecagons have twelve sides.
Calculating the perimeter and area of these shapes can be more difficult. Sometimes you have to come up with ways of doing it yourself. When you come across an equilateral polygon, you can of course multiply one of the sides by the number of sides of the shape. In other cases, you may need to find some dimensions yourself. Keep your eyes peeled for equivalences, and the problems cannot be that difficult.
When calculating the area of these shapes, there are two main ways of doing so: dissecting and filling. With dissecting, you cut up the figure into many pieces, such as parallelograms, squares and triangles. Then you can simply add up all those areas to find out the total. With filling, you add extra bits to shapes so as to make it look like the shapes you usually come across with. For example, when you don't known the altitude of a triangle, you can put three surrounding triangles around it. Then you can calculate the area of the rectangle formed and the surrounding triangles, thereby finding the area of the triangle.
Circles and other plane figures
Apart from polygons, there are other shapes that have wavy sides, round corners or other peculiarities that disqualify them as polygons. Among them, the most famous are the circle, the ellipse, and the semicircle. These shapes are different from polygons, and have their special formulae that you must learn by heart. Let's start with the most basic: the circle.
|The corresponding material in Euclid's elements can be found on page 27 of Book I, Definition(s) 15-17 in Issac Todhunter's 1872 translation, The Elements of Euclid for the Use of Schools and Colleges.|
Circles are shapes with infinite loci around its centre. Its perimeter is called the circumference. The line running from one side of the circle, through the centre and to the other side is called the diameter. The line running from the centre to any point on the circumference is called the radius. Any other line running from one point of the circumference to another is called a chord. An arc is any part of the circumference.
For thousands of years, mathematicians have been trying to find out the relationship between the circumference and the diameter. When we divide the circumference by the diameter, we get a number that is slightly larger than 3. That number is called π (spelt pi and pronounced pie). Supercomputers have discovered millions of digits of π, but you only need to remember that π is roughly 3.14 or 22/7. That is close enough. If you know the circumference of a circle, dividing that by π will result in the diameter; multiplying the diameter by π will result in the circumference. To find out the area of a circle, calculate πr2.
You don't really get to know much about ellipses and semicircles in elementary school. Ellipses look like ovals, except they have a stricter way of constructing that is more than a crushed circle. They have two 'centres' called foci. Semicircles are circles cut along the diameter, and if you draw a line from one end to a point on the circumference, then to another end, you always get a right angle. These two shapes are seldom taught in elementary school, and aside from knowing their names you don't need to study them.