Geometry/Right Triangles and Pythagorean Theorem

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Right triangles[edit]

Right triangles are triangles in which one of the interior angles is 90o. A 90o angle is called a right angle. Right triangles are sometimes called right-angled triangles. The other two interior angles are complementary, i.e. their sum equals 90o. Right triangles have special properties which make it easier to conceptualize and calculate their parameters in many cases.

The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are the legs. When using the Pythagorean Theorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.

Right triangle shows hyp legs.PNG

Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula:


Right triangle bh.PNGArea = \,(1/2) b h \,


This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b × h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle.

Two right triangles in rectangle.PNG

Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45° angles as well as the 90° angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles (see Triangle), then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagorean theorem can be used for non right triangles. a2+b2=c2-2c

Pythagorean Theorem[edit]

For history regarding the Pythagorean Theorem, see Pythagorean theorem. The Pythagorean Theorem states that:

  • In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's take a right triangle as shown here and set c equal to the length of the hypotenuse and set a and b each equal to the lengths of the other two sides. Then the Pythagorean Theorem can be stated as this equation:

Pythagorean triangle.PNG  \quad  c^2 = a^2 + b^2

Using the Pythagorean Theorem, if the lengths of any two of the sides of a right triangle are known and it is known which side is the hypotenuse, then the length of the third side can be determined from the formula.

Sine, Cosine, and Tangent for Right Triangles[edit]

Sine, Cosine, and Tangent are all functions of an angle, which are useful in right triangle calculations. For an angle designated as θ, the sine function is abbreviated as sin θ, the cosine function is abbreviated as cos θ, and the tangent function is abbreviated as tan θ. For any
angle θ, sin θ, cos θ, and tan θ are each single determined values and if θ is a known value, sin θ, cos θ, and tan θ can be looked up in a table or found with a calculator. There is a table listing these function values at the end of this section. For an angle between listed values, the sine, cosine, or tangent of that angle can be estimated from the values in the table. Conversely, if a number is known to be the sine, cosine, or tangent of a angle, then such tables could be used in reverse to find (or estimate) the value of a corresponding angle.

These three functions are related to right triangles in the following ways:

In a right triangle,

  • the sine of a non-right angle equals the length of the leg opposite that angle divided by the length of the hypotenuse.


  • the cosine of a non-right angle equals the length of the leg adjacent to it divided by the length of the hypotenuse.


  • the tangent of a non-right angle equals the length of the leg opposite that angle divided by the length of the leg adjacent to it.


For any value of θ where cos θ ≠ 0,

\qquad \tan \theta  = \frac {\sin \theta } {\cos \theta }.


If one considers the diagram representing a right triangle with the two non-right angles θ1and θ2, and the side lengths a,b,c as shown here:

Right triangle.PNG

For the functions of angle θ1:

\sin \theta_1 = \frac {b} {c} \qquad \cos \theta_1 = \frac{a} {c} \qquad \tan \theta_1 = \frac {b} {a}


Analogously, for the functions of angle θ2:

\sin \theta_2 = \frac {a} {c} \qquad \cos \theta_2 = \frac {b} {c} \qquad \tan \theta_2 = \frac {a} {b}



Table of sine, cosine, and tangent for angles θ from 0 to 90°[edit]

θ in degrees θ in radians sin θ cos θ tan θ
0 0 0.0 1.0 0.0
1 0.017453293 0.01745240 0.9998477 0.017455065
2 0.034906585 0.034899497 0.99939083 0.034920769
3 0.052359878 0.052335956 0.99862953 0.052407779
4 0.06981317 0.069756474 0.99756405 0.069926812
5 0.087266463 0.087155743 0.9961947 0.087488664
6 0.10471976 0.10452846 0.9945219 0.10510424
7 0.12217305 0.12186934 0.99254615 0.12278456
8 0.13962634 0.1391731 0.99026807 0.14054083
9 0.15707963 0.15643447 0.98768834 0.15838444
10 0.17453293 0.17364818 0.98480775 0.17632698
11 0.19198622 0.190809 0.98162718 0.19438031
12 0.20943951 0.20791169 0.9781476 0.21255656
13 0.2268928 0.22495105 0.97437006 0.23086819
14 0.2443461 0.2419219 0.97029573 0.249328
15 0.26179939 0.25881905 0.96592583 0.26794919
16 0.27925268 0.27563736 0.9612617 0.28674539
17 0.29670597 0.2923717 0.95630476 0.30573068
18 0.31415927 0.30901699 0.95105652 0.3249197
19 0.33161256 0.32556815 0.94551858 0.34432761
20 0.34906585 0.34202014 0.93969262 0.36397023
21 0.36651914 0.35836795 0.93358043 0.38386404
22 0.38397244 0.37460659 0.92718385 0.40402623
23 0.40142573 0.39073113 0.92050485 0.42447482
24 0.41887902 0.40673664 0.91354546 0.44522869
25 0.43633231 0.42261826 0.90630779 0.46630766
26 0.45378561 0.43837115 0.89879405 0.48773259
27 0.4712389 0.4539905 0.89100652 0.50952545
28 0.48869219 0.46947156 0.88294759 0.53170943
29 0.50614548 0.48480962 0.87461971 0.55430905
30 0.52359878 0.5 0.8660254 0.57735027
31 0.54105207 0.51503807 0.8571673 0.60086062
32 0.55850536 0.52991926 0.8480481 0.62486935
33 0.57595865 0.54463904 0.83867057 0.64940759
34 0.59341195 0.5591929 0.82903757 0.67450852
35 0.61086524 0.57357644 0.81915204 0.70020754
36 0.62831853 0.58778525 0.80901699 0.72654253
37 0.64577182 0.60181502 0.79863551 0.75355405
38 0.66322512 0.61566148 0.78801075 0.78128563
39 0.68067841 0.62932039 0.77714596 0.80978403
40 0.6981317 0.64278761 0.76604444 0.83909963
41 0.71558499 0.65605903 0.75470958 0.86928674
42 0.73303829 0.66913061 0.74314483 0.90040404
43 0.75049158 0.68199836 0.7313537 0.93251509
44 0.76794487 0.69465837 0.7193398 0.96568877
45 0.78539816 0.70710678 0.70710678 1.0
46 0.80285146 0.7193398 0.69465837 1.03553031
47 0.82030475 0.7313537 0.68199836 1.07236871
48 0.83775804 0.74314483 0.66913061 1.11061251
49 0.85521133 0.75470958 0.65605903 1.15036841
50 0.87266463 0.76604444 0.64278761 1.19175359
51 0.89011792 0.77714596 0.62932039 1.23489716
52 0.90757121 0.78801075 0.61566148 1.27994163
53 0.9250245 0.79863551 0.60181502 1.32704482
54 0.9424778 0.80901699 0.58778525 1.37638192
55 0.95993109 0.81915204 0.57357644 1.42814801
56 0.97738438 0.82903757 0.5591929 1.48256097
57 0.99483767 0.82367057 0.54463904 1.53986496
58 1.01229097 0.8480481 0.52991926 1.60033453
59 1.02974426 0.8571673 0.51503807 1.66427948
60 1.04719755 0.8660254 0.5 1.73205081
61 1.06465084 0.87461971 0.48480962 1.80404776
62 1.08210414 0.88294759 0.46947156 1.88072647
63 1.09955743 0.89100652 0.4539905 1.96261051
64 1.11701072 0.89879405 0.43837115 2.05030384
65 1.13446401 0.90630779 0.42261826 2.14450692
66 1.15191731 0.91354546 0.40673664 2.24603677
67 1.1693706 0.92050485 0.39073113 2.35585237
68 1.18682389 0.92718385 0.37460659 2.47508685
69 1.20427718 0.93358043 0.35836795 2.60508906
70 1.22173048 0.93969262 0.34202014 2.74747742
71 1.23918377 0.94551858 0.32556815 2.90421088
72 1.25663706 0.95105652 0.30901699 3.07768354
73 1.27409035 0.95630476 0.2923717 3.27085262
74 1.29154365 0.9612617 0.27563736 3.48741444
75 1.30899694 0.96592583 0.25881905 3.73205081
76 1.32645023 0.97029573 0.2419219 4.01078093
77 1.34390352 0.97437006 0.22495105 4.33147587
78 1.36135682 0.9781476 0.20791169 4.70463011
79 1.37881011 0.98162718 0.190809 5.14455402
80 1.3962634 0.98480775 0.17364818 5.67128182
81 1.41371669 0.98768834 0.15643447 6.31375151
82 1.43116999 0.99026807 0.1391731 7.11536972
83 1.44862328 0.99254615 0.12186934 8.14434643
84 1.46607657 0.9945219 0.10452846 9.51436445
85 1.48352986 0.9961947 0.087155743 11.4300523
86 1.50098316 0.99756405 0.069756474 14.3006663
87 1.51843645 0.99862953 0.052335956 19.0811367
88 1.53588974 0.99939083 0.034899497 28.6362533
89 1.55334303 0.9998477 0.01745240 57.2899616
90 1.57079633 1.0 0.0 not defined


General rules for important angles:  sin 45^o = cos 45^o = \frac{1}{\sqrt{2}}

 tan 45^o = 1

 sin 30^o = \frac{1}{2} = cos 60^o

 cos 30^o = \frac{\sqrt{3}}{2} = sin 60^o