Longest side of a right-angled triangle, the side opposite of the right angle
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A right-angled triangle and its hypotenuse
In geometry, a hypotenuse is the side of a right triangle opposite the right angle.[1] It is the longest side of any such triangle; the two other shorter sides of such a triangle are called catheti or legs. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as , where a is the length of one leg, b is the length of another leg, and c is the length of the hypotenuse.[2]
For example, if one of the legs of a right angle has a length of 3 and the other has a length of 4, then their squares add up to 25 = 9 + 16 = 3 × 3 + 4 × 4. Since 25 is the square of the hypotenuse, the length of the hypotenuse is the square root of 25, that is, 5. In other words, if and , then .
^"Triangle (geometry)" . Encyclopædia Britannica. Vol. 27 (11th ed.). 1911. p. 258. ...Also a right-angled triangle has one angle a right angle, the side opposite this angle being called the hypotenuse;...
^Jr, Jesse Moland (August 2009). I Hate Trig!: A Practical Guide to Understanding Trigonometry. Jesse Moland. p. 1. ISBN 978-1-4486-4707-1.
In geometry, a hypotenuse is the side of a right triangle opposite the right angle. It is the longest side of any such triangle; the two other shorter...
turn or 90 degrees). The side opposite to the right angle is called the hypotenuse (side c {\displaystyle c} in the figure). The sides adjacent to the right...
right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of...
side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle θ {\displaystyle...
of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles...
the angle to the hypotenuse. sin A = opposite hypotenuse = a h . {\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}...
being the hypotenuse of the prior triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this...
a h {\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}={\frac {a}{h}}} cos θ = a d j a c e n t h y p o t e n u s e = b...
outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse. It is also possible to compute the distance for points...
relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean...
right angle". The side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as "the...
13x5 if it were, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope, even though...
ratio of the hypotenuse to the sum of the legs, namely √2/2.: p. 282, p. 358 and the greatest ratio of the altitude from the hypotenuse to the sum of...
the hypotenuse. The toes of the weld are essentially the edges or the points of the hypotenuse. The face of the weld is the outer visual or hypotenuse that...
right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ. The elementary definitions of the sine and...
the hypotenuse have the sum of squares of inverses of the integer legs equal to the square of the inverse of the integer altitude from the hypotenuse. Pythagorean...
and hypotenuse a 2 {\displaystyle a^{2}} also would have integer sides including a square leg ( b 2 {\displaystyle b^{2}} ) and a square hypotenuse ( a...
which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents...
chosen to be a whole number (for ease of later calculations) and forms the hypotenuse of a triangle when in use. When a sine bar is placed on a level surface...
the angle divided by the hyperbolic sine of the hypotenuse. sin A = sinh(opposite) sinh(hypotenuse) = sinh a sinh c . {\displaystyle \sin A={\frac...
with respect to that of the radar antenna. The slant range (1) is the hypotenuse of the triangle represented by the altitude of the aircraft and the distance...
the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; 5 {\displaystyle {\sqrt {5}}} is the hypotenuse of...
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