Mathematics: Geometry - Two-Dimensional Shapes

Polygons
A polygon is a closed, two-dimensional figure with three or more straight line segments called sides.
The point at which two sides of a polygon intersect is called the vertex.

In a polygon, the number of sides is always equal to the number of vertices. A polygon with all sides congruent and all angles equal is called a regular polygon. Common polygons are:

More generally, an n-gon is a polygon that has n angles and n sides.
 

The sum of the interior angles of an n-sided polygon is
. For example, in a triangle
. So the sum of the interior angles is
. In a quadrilateral,
, and the sum of the angles is
.

Apothem and Radius
A line segment from the center of a polygon that is perpendicular to a side of the polygon is called the apothem. A line segment from the center of a polygon to a vertex of the polygon is called a radius. In a regular polygon, the apothem can be used to find the area of the polygon using the formula
is the apothem, and <i>p</i> is the perimeter.<br><img data-cke-saved-src=" />
A diagonal is a line segment that joins two non-adjacent vertices of a polygon. The number of diagonals a polygon has can be found by using the formula:

Note that n is the number of sides in the polygon. This formula works for all polygons, not just regular polygons.

Convex and Concave Polygons
A convex polygon is a polygon whose diagonals all lie within the interior of the polygon. A concave polygon is a polygon with a least one diagonal that is outside the polygon.
In the diagram below, quadrilateral ABCD is concave because diagonal

lies outside the polygon and quadrilateral EFGH is convex because both diagonals lie inside the polygon
Concave
Convex



Congruence and Similarity
Congruent figures are geometric figures that have the same size and shape. All corresponding angles are equal, and all corresponding sides are equal. Congruence is indicated by the symbol
.

Similar figures are geometric figures that have the same shape, but do not necessarily have the same size.
All corresponding angles are equal, and all corresponding sides are proportional, but they do not have to be equal. It is indicated by the symbol
.

Note that all congruent figures are also similar, but not all similar figures are congruent.

Line of Symmetry
A line that divides a figure or object into congruent parts is called a line of symmetry. An object may have no lines of symmetry, one line of symmetry, or multiple (i.e., more than one) lines of symmetry.

None -

One -

Multiple -


Triangles
A triangle is a three-sided figure with the sum of its interior angles being 
perimeter of any triangle is found by summing the three side lengths; <br><img data-cke-saved-src=" />.
For an equilateral triangle, this is the same as
, where a is any side length, since all three sides are the same length.

The area of any triangle can be found by taking half the product of one side length referred to as the base, often given the variable b and the perpendicular distance from that side to the opposite vertex called the altitude or height and given the variable h.
In equation form that is
is the semiperimeter: <br><img src=" />a, b, and c are the lengths of the three sides. Special cases include isosceles triangles:
is the unique side and <i>a</i> is the length of one of the two congruent sides, and equilateral triangles: <br><img src=" />a is the length of a side.

Parts of a Triangle

An altitude of a triangle is a line segment drawn from one vertex perpendicular to the opposite side. In the diagram below,
,
, and

are altitudes. The length of an altitude is also called the height of the triangle.

The three altitudes in a triangle are always concurrent. The point of concurrency of the altitudes of a triangle, O, is called the orthocenter.
Note that in an obtuse triangle, the orthocenter will be outside the triangle, and in a right triangle, the orthocenter is the vertex of the right angle.
A median of a triangle is a line segment drawn from one vertex to the midpoint of the opposite side. In the diagram below,
,
, and

are medians.

This is not the same as the altitude, except the altitude to the base of an isosceles triangle and all three altitudes of an equilateral triangle. The point of concurrency of the medians of a triangle, T, is called the centroid.

This is the same point as the orthocenter only in an equilateral triangle.

Unlike the orthocenter, the centroid is always inside the triangle. The centroid can also be considered the exact center of the triangle.
Any shape triangle can be perfectly balanced on a tip placed at the centroid.
The centroid is also the point that is two-thirds the distance from the vertex to the opposite side.





Quadrilaterals
A quadrilateral is a closed two-dimensional geometric figure that has four straight sides. The sum of the interior angles of any quadrilateral is
.

Kite
A kite is a quadrilateral with two pairs of adjacent sides that are congruent. A result of this is perpendicular diagonals. A kite can be concave or convex and has one line of symmetry.

Trapezoid
Trapezoid: A trapezoid is defined as a quadrilateral that has at least one pair of parallel sides.
There are no rules for the second pair of sides. So there are no rules for the diagonals and no lines of symmetry for a trapezoid.

The area of a trapezoid is found by the formula
is the height (segment joining and perpendicular to the parallel bases), and <i>b</i>₁ and <i>b</i>₂ are the two parallel sides (bases). Do not use one of the other two sides as the height unless that side is also perpendicular to the parallel bases.<br> The perimeter of a trapezoid is found by the formula <br><img src=" />a, b1, c, and b2 are the four sides of the trapezoid.


Isosceles trapezoid: A trapezoid with equal base angles. This gives rise to other properties including: the two nonparallel sides have the same length, the two non-base angles are also equal, and there is one line of symmetry through the midpoints of the parallel sides.

Parallelogram
Parallelogram: A quadrilateral that has two pairs of opposite parallel sides. As such it is a special type of trapezoid. The sides that are parallel are also congruent. The opposite interior angles are always congruent, and the consecutive interior angles are supplementary. The diagonals of a parallelogram divide each other.
Each diagonal divides the parallelogram into two congruent triangles. A parallelogram has no line of symmetry, but does have 180-degree rotational symmetry about the midpoint.

The area of a parallelogram is found by the formula
, where b is the length of the base, and h is the height. Note that the base and height correspond to the length and width in a rectangle, so this formula would apply to rectangles as well. Do not confuse the height of a parallelogram with the length of the second side. The two are only the same measure in the case of a rectangle. parallelogram is found by the formula

or
, where a and b are the lengths of the two sides.

Rectangle
Rectangle: A quadrilateral with four right angles. All rectangles are parallelograms and trapezoids, but not all parallelograms or trapezoids are rectangles. The diagonals of a rectangle are congruent. Rectangles have 2 lines of symmetry (through each pair of opposing midpoints) and 180-degree rotational symmetry about the midpoint.

The area of a rectangle is found by the formula
, where A is the area of the rectangle, l is the length (usually considered to be the longer side) and w is the width (usually considered to be the shorter side). The numbers for l and w are interchangeable. rectangle is found by the formula

or
, where l is the length, and w is the width. It may be easier to add the length and width first and then double the result, as in the second formula.

Rhombus
Rhombus: A quadrilateral with four congruent sides. All rhombuses are parallelograms and kites; thus, they inherit all the properties of both types of quadrilaterals.
The diagonals of a rhombus are perpendicular to each other. Rhombi have 2 lines of symmetry (along each of the diagonals) and 180-degree rotational symmetry.

The area of a rhombus is half the product of the diagonals:

and the perimeter of a rhombus is:



Square
Square: A quadrilateral with four right angles and four congruent sides. Squares satisfy the criteria of all other types of quadrilaterals. The diagonals of a square are congruent and perpendicular to each other. Squares have 4 lines of symmetry (through each pair of opposing midpoints and along each of the diagonals) as well as 90-degree rotational symmetry about the midpoint.

The area of a square is found by using the formula
is the length of one side. The perimeter of a square is found by using the formula <br><img src=" />s is the length of one side. Because all four sides are equal in a square, it is faster to multiply the length of one side by 4 than to add the same number four times. You could use the formulas for rectangles and get the same answer.

Hierarchy of Quadrilaterals
The hierarchy of quadrilaterals can be shown as follows:


Circles
The center of a circle is the single point from which every point on the circle is equidistant.
The radius is a line segment that joins the center of the circle and any one point on the circle. All radii of a circle are equal. Circles that have the same center, but not the same length of radii are concentric. The diameter is a line segment that passes through the center of the circle and has both endpoints on the circle. The length of the diameter is exactly twice the length of the radius. Point O in the diagram below is the center of the circle, segments
,
, and

are radii, and segment
is a diameter.


The area of a circle is found by the formula
is the length of the radius. If the diameter of the circle is given, remember to divide it in half to get the length of the radius before proceeding.<br> The circumference of a circle is found by the formula <br><img src=" />r is the radius. Again, remember to convert the diameter if you are given that measure rather than the radius.

Inscribed and Circumscribed Figures
These terms can both be used to describe a given arrangement of figures, depending on perspective. If each of the vertices of figure A lie on figure B, then it can be said that figure A is inscribed in figure B, but it can also be said that figure B is circumscribed about figure A.
The following table and examples help to illustrate the concept. Note that the figures cannot both be circles, as they would be completely overlapping and neither would be inscribed or circumscribed.





Each of the vertices of a pentagon lie on a circle inscribed in the circle circumscribed about the pentagon

 

P1. Find the area and perimeter of the following quadrilaterals:
(a) A square with side length 2.5 cm.
(b) A parallelogram with height 3 m, base 4 m, and other side 6 m.
(c) A rhombus with diagonals 15 in and 20 in.
P2. Calculate the area of a triangle with side lengths of 7 ft, 8 ft, and 9 ft.
P3. Square ABCD is inscribed in a circle with radius 20 m. What is the area of the part of the circle outside of the square?

P1. (a)
;

(b)  
(c)  ;


P2. Given only side lengths, we can use the semi perimeter to the find the area based on the formula,
is the semiperimeter, <br><img data-cke-saved-src=" />:

P3. Begin by drawing a diagram of the situation, where we want to find the shaded area:





The area of the square is
, so the area we want to find is:
. Since the inscribed figure is a square, the triangle BCO is a 45-45-90 right triangle.
Now we can find
. So, the shaded area is: