pythagorean theorem - a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides

Formula: a^{2}+ b^{2}= c^{2}

A 13ft ladder leans against the side of a house. The bottom of the ladder is 10ft from the side of t

A 13ft ladder leans against the side of a house. The bottom of the ladder is 10ft from the side of the house. How high is the top of the ladder from the ground? If necessary, round your answer to the nearest tenth.
We have a right triangle. Hypotenuse = 13, one leg = 10.
We use our [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=10&hypinput=13&pl=Solve+Missing+Side']Pythagorean theorem Calculator to solve for the other leg[/URL]:
s = [B]8.3066[/B]

A 50-foot pole and a 70-foot pole are 30 feet apart. If you were to run a line between the tops of t

A 50-foot pole and a 70-foot pole are 30 feet apart. If you were to run a line between the tops of the two poles, what is the minimum length of cord you would need?
The difference between the 70 foot and 50 foot pole is:
70 - 50 = 20 foot height difference.
So we have a right triangle, with a height of 20, base of 30. We want to know the hypotenuse.
Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=20&side2input=30&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator to solve for hypotenuse[/URL], we get:
hypotenuse = [B]36.06 feet[/B]

A ladder 25 feet long is leaning against a wall. If the base of the ladder is 7 feet from the wall,

A ladder 25 feet long is leaning against a wall. If the base of the ladder is 7 feet from the wall, how high up the wall does the ladder reach?
We have a right triangle, where the ladder is the hypotenuse, and we want the measurement of one leg.
Set up the pythagorean theorem with these given items using our P[URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=7&hypinput=25&pl=Solve+Missing+Side']ythagorean Theorem Calculator[/URL].
We get Side 1 = [B]24 feet.[/B]

A ladder is 25 ft long. The ladder needs to reach to a window that is 24 ft above the ground. How fa

A ladder is 25 ft long. The ladder needs to reach to a window that is 24 ft above the ground. How far away from the building should the bottom of the ladder be placed?
We have a right triangle, where the ladder is the hypotenuse, and the window side is one side.
Using our right triangle and the [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=24&hypinput=25&pl=Solve+Missing+Side']pythagorean theorem calculator[/URL], we get a length of [B]7 ft [/B]for the ladder bottom from the wall.

A young dad, who was a star football player in college, set up a miniature football field for his fi

A young dad, who was a star football player in college, set up a miniature football field for his five-year-old young daughter, who was already displaying an unusual talent for place-kicking. At each end of the mini-field, he set up goal posts so she could practice kicking extra points and field goals. He was very careful to ensure the goalposts were each straight up and down and that the crossbars were level. On each set, the crossbar was six feet long, and a string from the top of each goalpost to the midpoint between them on the ground measured five feet. How tall were the goalposts? How do you know this to be true?
The center of each crossbar is 3 feet from each goalpost. We get this by taking half of 6, since midpoint means halfway.
Imagine a third post midway between the two goal posts. It has the same height as the two goalposts.
From the center post, the string from the top of a goalpost to the base of the center post, and half the crossbar form and right triangle with hypotenuse 5 feet and one leg 3 feet.
[URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=3&hypinput=5&pl=Solve+Missing+Side']Using the Pythagorean Theorem[/URL], the other leg -- the height of each post -- is 4 feet.

Gigi’s family left their house and drove 14 miles south to a gas station and then 48 miles east to a

Gigi’s family left their house and drove 14 miles south to a gas station and then 48 miles east to a water park. How much shorter would their trip to the water park have been if they hadn’t stopped at the gas station and had driven along the diagonal path instead?
[IMG]https://mathcelebrity.com/community/data/attachments/0/pythag-diagonal.jpg[/IMG]
Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=14&side2input=48&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we see the diagonal route would be:
50 miles
The original trip distance was:
Original Trip Distance = 14 + 48
Original Trip Distance = 62 miles
Diagonal Trip was 50 miles, so the difference is:
Difference = Original Trip Distance - Diagonal Distance
Difference = 62 - 50
Difference = [B]12 miles[/B]

Given the rectangular prism below, if AB = 6 in., AD = 8 in. and BF = 24, find the length of FD.

Given the rectangular prism below, if AB = 6 in., AD = 8 in. and BF = 24, find the length of FD.
[IMG]http://www.mathcelebrity.com/images/math_problem_library_129.png[/IMG]
If AB = 6 and AD = 8, by the Pythagorean theorem, we have BD = 10 from our [URL='http://www.mathcelebrity.com/pythag.php?side1input=6&side2input=8&hypinput=&pl=Solve+Missing+Side']Pythagorean Theorem[/URL] Calculator
Using that, we have another right triangle which we can use the [URL='http://www.mathcelebrity.com/pythag.php?side1input=10&side2input=24&hypinput=&pl=Solve+Missing+Side']pythagorean theorem[/URL] calculator to get [B]FD = 26[/B]

Pythagorean Theorem

Free Pythagorean Theorem Calculator - Figures out based on user entry the missing side or missing hypotenuse of a right triangle. In addition, the calculator shows the proof of the Pythagorean Theorem and then determines by numerical evaluation if the 2 sides and hypotenuse you entered are a right triangle using the Pythagorean Theorem

Pythagorean Theorem Trig Proofs

Free Pythagorean Theorem Trig Proofs Calculator - Shows the proof of 3 pythagorean theorem related identities using the angle θ:

Sin^{2}(θ) + Cos^{2}(θ) = 1

Tan^{2}(θ) + 1 = Sec^{2}(θ)

Sin(θ)/Cos(θ) = Tan(θ)

Sin

Tan

Sin(θ)/Cos(θ) = Tan(θ)

Right Triangles

Free Right Triangles Calculator - This solves for all the pieces of a right triangle based on given inputs using items like the sin ratio, cosine ratio, tangent ratio, and the Pythagorean Theorem as well as the inradius.

Running from the top of a flagpole to a hook in the ground there is a rope that is 9 meters long. If

Running from the top of a flagpole to a hook in the ground there is a rope that is 9 meters long. If the hook is 4 meters from the base of the flagpole, how tall is the flagpole?
We have a right triangle, with hypotenuse of 9 and side of 4.
[URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=4&hypinput=9&pl=Solve+Missing+Side']Using our Pythagorean Theorem calculator[/URL], we get a flagpole height of [B]8.063[/B].

Sam leaves school to go home. He walks 10 blocks North and then 8 blocks west. How far is John from

Sam leaves school to go home. He walks 10 blocks North and then 8 blocks west. How far is John from the school?
Sam walked at a right angle. His distance from home to school is the hypotenuse.
Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=8&side2input=10&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we get:
[B]12.806 blocks[/B]

The distance between consecutive bases is 90 feet. An outfielder catches the ball on the third base

The distance between consecutive bases is 90 feet. An outfielder catches the ball on the third base line about 40 feet behind third base. How far would the outfielder have to throw the ball to first base?
We have a right triangle. From home base to third base is 90 feet. We add another 40 feet to the outfielder behind third base to get: 90 + 40 = 130
The distance from home to first is 90 feet.
Our hypotenuse is the distance from the outfielder to first base.
[URL='https://www.mathcelebrity.com/pythag.php?side1input=130&side2input=90&hypinput=&pl=Solve+Missing+Side']Using our Pythagorean theorem calculator[/URL], we get:
d = [B]158.11 feet[/B]