Geometry - How to find the length of a chord

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Explanation

Draw a segment perpendicular to the chord from the center, and this line will bisect the chord. Setting up the Pythagorean Theorem with the radius as the hypotenuse and the distance as one of the legs, we solve for the other leg.

$10^2-6^2=64 \rightarrow \sqrt{64}=8$

Since this leg is half of the chord, the total chord length is 2 times that, or 16.

Find the length of the chord in the figure below.

Explanation

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{6^2-3^2}=\sqrt{27}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=2\sqrt{27}=10.39$

Make sure to round to places after the decimal.

Find the length of the chord in the figure below.

Explanation

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{9^2-8^2}=\sqrt{17}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=2\sqrt{17}=8.25$

Make sure to round to places after the decimal.

Find the length of the chord in the figure below.

Explanation

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{7^2-5^2}=\sqrt{24}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=2\sqrt{24}=9.80$

Make sure to round to places after the decimal.

Find the length of chord .

Explanation

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{10^2-6^2}=2\sqrt{64}=16$

If a chord is units away from the center of a circle, and the radius is , what is the length of that chord?

Explanation

$5^2-4^2=9 \rightarrow \sqrt{9}=3$

Since this leg is half of the chord, the total chord length is 2 times that, or 6.

Find the length of the chord .

Explanation

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{24^2-12^2}=2\sqrt{432}=41.57$

Make sure to round to two places after the decimal.

Find the length of the chord in the figure below.

Explanation

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$

$\text{Half Chord}=\sqrt{11^2-8^2}=\sqrt{57}$

Multiply this by to find the length of the entire chord.

$\text{Length of chord}=2\sqrt{57}=15.10$

Make sure to round to places after the decimal.

Find the length of chord .

Explanation

Recall that the perpendicular distance from the center of the circle to the chord will bisect the chord itself.

Thus, we can use the Pythagorean Theorem to find the length of the chord.

$\text{Radius}^2=\text{distance}^2+\text{half chord}^2$

Rearrange the equation to solve for the length of half the chord.

$\text{Half Chord}=\sqrt{\text{Radius}^2-\text{Distance}^2}$

Now, multiply this value by two to find the length of the entire chord.

$\text{Chord}=2\sqrt{\text{Radius}^2-\text{Distance}^2}$

Plug in the given radius and distance to find the length of the chord.

$\text{Chord}=2\sqrt{9^2-4^2}=2\sqrt{65}=16.12$

Make sure to round to two places after the decimal.

Find the area of the shaded region in the figure below.

Explanation

When a chord is intercepted by a perpendicular line segment originating at the center of the circle, the chord is bisected, or cut in half. From the figure, you should notice that the base of the triangle is also the chord of the circle.

Thus, we can use the Pythagorean Theorem to find the half of the chord that acts as the other leg of the right triangle that is created.

$\text{Half Chord}=\sqrt{\text{radius}^2-\text{leg}^2}$