How to find the length of the diagonal of a square - Math
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Which is the greater quantity?
(a) The length of a diagonal of a square with sidelength 10 inches
(b) The hypotenuse of an isosceles right triangle with legs 10 inches each
Which is the greater quantity?
(a) The length of a diagonal of a square with sidelength 10 inches
(b) The hypotenuse of an isosceles right triangle with legs 10 inches each
A diagonal of a square cuts the square into two isosceles right triangles, of which the diagonal is the common hypotenuse. Therefore, each figure is the hypotenuse of an isosceles right triangle with legs 10 inches, making them equal in length.
A diagonal of a square cuts the square into two isosceles right triangles, of which the diagonal is the common hypotenuse. Therefore, each figure is the hypotenuse of an isosceles right triangle with legs 10 inches, making them equal in length.
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The track at Peter Stuyvesant High School is a perfect square, as seen above, with sides of length 600 feet and a diagonal connecting two of the corners.
Les begins at Point A, takes the diagonal path directly to Point B, then runs counterclockwise around the square track twice. He then takes the diagonal from Point B back to Point A. Which of the following is closest to the distance he runs?
A hint: 
The track at Peter Stuyvesant High School is a perfect square, as seen above, with sides of length 600 feet and a diagonal connecting two of the corners.
Les begins at Point A, takes the diagonal path directly to Point B, then runs counterclockwise around the square track twice. He then takes the diagonal from Point B back to Point A. Which of the following is closest to the distance he runs?
A hint:
The diagonal of a square has length 
, or about 1.414, times the length of a side, which here is 600 feet; this makes the diagonal path about
feet long.
Les runs around the square track twice, meaning that he runs the length of one side eight times; he also runs the length of the diagonal twice, This is a total of about
feet.
Divide by 5,280 to convert to miles:
Of the given responses, 
miles comes closest to the correct distance.
The diagonal of a square has length
Les runs around the square track twice, meaning that he runs the length of one side eight times; he also runs the length of the diagonal twice, This is a total of about
Divide by 5,280 to convert to miles:
Of the given responses,
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The track at Franklin Pierce High School is a perfect square, as seen above, with sides of length 700 feet and a diagonal path connecting Points A and C.
Ellen wants to run three miles. Her plan is to begin at Point A, run along the diagonal path, run clockwise around the square track once, run along the diagonal path, run clockwise around the square track once, then repeat this pattern until she has run three miles. Where will she be when she is done?
A hint: 
The track at Franklin Pierce High School is a perfect square, as seen above, with sides of length 700 feet and a diagonal path connecting Points A and C.
Ellen wants to run three miles. Her plan is to begin at Point A, run along the diagonal path, run clockwise around the square track once, run along the diagonal path, run clockwise around the square track once, then repeat this pattern until she has run three miles. Where will she be when she is done?
A hint:
The diagonal of a square has length 
, or about 1.414, times the length of a side, which here is 700 feet; this makes the diagonal path about
feet long.
We will call one complete circuit one running of the diagonal, which is 990 feet long, and one running around the square; the completion of one complete circuit amounts to running a distance of
feet.
Ellen seeks to run three miles, or
feet, which, divided by 3,790 feet, is about:
,
or four complete circuits and 0.17 of a fifth.
After four complete circuits, Ellen is backat Point A. She has yet to run
feet.
She will now run along the diagonal from Point A to Point C, but since the diagonal has length 990 feet, which is greater than 629 feet, she will finish running three miles when she is on this diagonal path.
The diagonal of a square has length
We will call one complete circuit one running of the diagonal, which is 990 feet long, and one running around the square; the completion of one complete circuit amounts to running a distance of
Ellen seeks to run three miles, or
or four complete circuits and 0.17 of a fifth.
After four complete circuits, Ellen is backat Point A. She has yet to run
She will now run along the diagonal from Point A to Point C, but since the diagonal has length 990 feet, which is greater than 629 feet, she will finish running three miles when she is on this diagonal path.
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The track at Grant High School is a perfect square, as seen above, with sides of length 600 feet and a diagonal path connecting two of the corners.
Kenny begins at Point A, runs the path to Point C, and proceeds to run counterclockwise around the square track one complete time. He then runs again along the diagonal path from Point C to Point A.
Which is the greater quantity?
(a) The length of Kenny's run
(b) One mile
A hint: 
The track at Grant High School is a perfect square, as seen above, with sides of length 600 feet and a diagonal path connecting two of the corners.
Kenny begins at Point A, runs the path to Point C, and proceeds to run counterclockwise around the square track one complete time. He then runs again along the diagonal path from Point C to Point A.
Which is the greater quantity?
(a) The length of Kenny's run
(b) One mile
A hint:
The diagonal of a square has length , or about 1.414, times the length of a side, which here is 600 feet; this makes the diagonal path about
feet long.
Kenny runs along this path twice, and he runs along the entire perimeter of the square path, so his run is about
feet. Since one mile is equal to 5,280 feet, the greater quantity is (b).
The diagonal of a square has length
Kenny runs along this path twice, and he runs along the entire perimeter of the square path, so his run is about
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The perimeter of a square is 48. What is the length of its diagonal?
The perimeter of a square is 48. What is the length of its diagonal?
Perimeter = side * 4
48 = side * 4
Side = 12
We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.
Therefore, we can use the Pythagorean Theorem to solve for the diagonal:
Perimeter = side * 4
48 = side * 4
Side = 12
We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.
Therefore, we can use the Pythagorean Theorem to solve for the diagonal:
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A square lot has an area of 1,200 square meters. To the nearest meter, how far is it from one corner to the opposite corner?
A square lot has an area of 1,200 square meters. To the nearest meter, how far is it from one corner to the opposite corner?
A square is also a rhombus, so its area can be calculated as one half the product of its diagonals:
,
where 
is the common diagonal length.
Since 
, 
.
The distance between opposite corners is about 49 meters.
A square is also a rhombus, so its area can be calculated as one half the product of its diagonals:
where
Since
The distance between opposite corners is about 49 meters.
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Find the length of the square's diagonal.
Find the length of the square's diagonal.
The diagonal line cuts the square into two equal triangles. Their hypotenuse is the diagonal of the square, so we can solve for the hypotenuse.
We need to use the Pythagorean Theorem: 
, where a and b are the legs and c is the hypotenuse.
The two legs have lengths of 8. Plug this in and solve for c:
The diagonal line cuts the square into two equal triangles. Their hypotenuse is the diagonal of the square, so we can solve for the hypotenuse.
We need to use the Pythagorean Theorem:
The two legs have lengths of 8. Plug this in and solve for c:
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Side 
in the square below has a length of 12. What is the length of the diagonal 
?
Side
Diagonal 
forms a triangle with adjacent sides 
. Since this is a square we know this is a right triangle and we can use the Pythagorean Theorem to determine the length of 
. Sides of length 
form each of the legs and 
is the hypotenuse. So the equation looks like this:
Solve for 
We can simplify this to
Diagonal
Solve for
We can simplify this to
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A man wants to build a not-quite-regulation softball field on his property and finds that he only has enough room to make the distance between home plate and first base 44 feet. How far (nearest foot) will it be from home plate to second base, assuming he builds it to that specification?
(Note: the four bases are the vertices of a perfect square, with the bases called home plate, first base, second base, third base, in that order).
A man wants to build a not-quite-regulation softball field on his property and finds that he only has enough room to make the distance between home plate and first base 44 feet. How far (nearest foot) will it be from home plate to second base, assuming he builds it to that specification?
(Note: the four bases are the vertices of a perfect square, with the bases called home plate, first base, second base, third base, in that order).
The path from home plate to first base is a side of a perfect square; the path from home plate to second base is a diagonal. As two sides and a diagonal form a 
triangle, the diagonal measures 
as long as a side.
The distance to second base from home is 
times the distance to first base:
The path from home plate to first base is a side of a perfect square; the path from home plate to second base is a diagonal. As two sides and a diagonal form a
The distance to second base from home is
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Find the length of the diagonal of a square that has side lengths of 
cm.
Find the length of the diagonal of a square that has side lengths of
You can do this problem in two different ways that lead to the final answer:
1. Pythagorean Theorem
2. Special Triangles (45-45-90)
1. For the first idea, use the Pythagorean Theorem: 
, where a and b are the side lengths of the square and c is the length of the diagonal.
2. If you know that ALL squares can be made into two special right triangles such that their angles are 45-45-90, then there's a formula you could use:
Let's say that your side length of the square is "a". Then the diagonal of the square (or the hypotenuse of the right triangle) will be 
.
So using this with a=4:
You can do this problem in two different ways that lead to the final answer:
1. Pythagorean Theorem
2. Special Triangles (45-45-90)
1. For the first idea, use the Pythagorean Theorem:
2. If you know that ALL squares can be made into two special right triangles such that their angles are 45-45-90, then there's a formula you could use:
Let's say that your side length of the square is "a". Then the diagonal of the square (or the hypotenuse of the right triangle) will be
So using this with a=4:
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The perimeter of a square is 
units. How many units long is the diagonal of the square?
The perimeter of a square is
From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,
Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.
From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,
Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.
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Find the length of the diagonal of the square with a side length of 
.
Find the length of the diagonal of the square with a side length of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square with side lengths of 
.
Find the length of the diagonal of a square with side lengths of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square with side lengths of 
.
Find the length of the diagonal of a square with side lengths of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square with side lengths of 
.
Find the length of the diagonal of a square with side lengths of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square with side lengths of 
.
Find the length of the diagonal of a square with side lengths of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square with a side length of 
.
Find the length of the diagonal of a square with a side length of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square that has a side length of 
.
Find the length of the diagonal of a square that has a side length of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square that has a side length of 
.
Find the length of the diagonal of a square that has a side length of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above
Find the length of the diagonal of a square with side lengths of 
.
Find the length of the diagonal of a square with side lengths of
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.
Use the Pythagorean Theorem to find the length of the diagonal.
For the square given in the question,
Compare your answer with the correct one above