Diameter and Chords - PSAT Math
Card 0 of 56
The circle above has a radius of 
, and the measure of 
is 
. What is the length of chord 
?
The circle above has a radius of
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To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the 
side of the model triangle.
Therefore, 
Because x is equal to half of the chord, the answer is 
.
To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the
Therefore,
Because x is equal to half of the chord, the answer is
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
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Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
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For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
The perimeter of a circle is 36 π. What is the diameter of the circle?
The perimeter of a circle is 36 π. What is the diameter of the circle?
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The perimeter of a circle = 2 πr = πd
Therefore d = 36
The perimeter of a circle = 2 πr = πd
Therefore d = 36
If the area of the circle touching the square in the picture above is 
, what is the closest value to the area of the square?
If the area of the circle touching the square in the picture above is
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Obtain the radius of the circle from the area.
Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be 
, 
, and 
. The radii of the circle (from the center to the corners of the square) will be 9. The hypotenuse (side of the square) must be 
.
The area of the square is then 
.
Obtain the radius of the circle from the area.
Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be
The area of the square is then
Note: Figure NOT drawn to scale.
In the above circle, the length of arc 
is 
, and 
. What is the diameter of the circle?
Note: Figure NOT drawn to scale.
In the above circle, the length of arc
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Call the diameter 
. Since 
, 
is 
of the circle, and 
is 
of a circle with circumference 
.
is 
in length, so
Call the diameter
Note: Figure NOT drawn to scale.
In the above circle, the length of arc 
is 10, and 
. Give the diameter of the circle. (Nearest tenth).
Note: Figure NOT drawn to scale.
In the above circle, the length of arc
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Call the diameter 
. Since 
, 
is 
of a circle with circumference 
. Since it is of length 10, the circumference of the circle is 5 times this, or 50. Therefore, set 
in the circumference formula:
Call the diameter
Let 
represent the area of a circle and 
represent its circumference. Which of the following equations expresses 
in terms of 
?
Let
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The formula for the area of a circle is 
, and the formula for circumference is 
. If we solve for C in terms of r, we get
.
We can then substitute this value of r into the formula for the area:
The formula for the area of a circle is
We can then substitute this value of r into the formula for the area:
The circle above has a radius of , and the measure of is . What is the length of chord ?
The circle above has a radius of
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To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the side of the model triangle.
Therefore,
Because x is equal to half of the chord, the answer is .
To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the
Therefore,
Because x is equal to half of the chord, the answer is
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
Tap to see back →
Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
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For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
The perimeter of a circle is 36 π. What is the diameter of the circle?
The perimeter of a circle is 36 π. What is the diameter of the circle?
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The perimeter of a circle = 2 πr = πd
Therefore d = 36
The perimeter of a circle = 2 πr = πd
Therefore d = 36
If the area of the circle touching the square in the picture above is , what is the closest value to the area of the square?
If the area of the circle touching the square in the picture above is
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Obtain the radius of the circle from the area.
Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be , , and . The radii of the circle (from the center to the corners of the square) will be 9. The hypotenuse (side of the square) must be .
The area of the square is then .
Obtain the radius of the circle from the area.
Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be
The area of the square is then
Note: Figure NOT drawn to scale.
In the above circle, the length of arc is , and . What is the diameter of the circle?
Note: Figure NOT drawn to scale.
In the above circle, the length of arc
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Call the diameter . Since , is of the circle, and is of a circle with circumference .
is in length, so
Call the diameter
Note: Figure NOT drawn to scale.
In the above circle, the length of arc is 10, and . Give the diameter of the circle. (Nearest tenth).
Note: Figure NOT drawn to scale.
In the above circle, the length of arc
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Call the diameter . Since , is of a circle with circumference . Since it is of length 10, the circumference of the circle is 5 times this, or 50. Therefore, set in the circumference formula:
Call the diameter
Let represent the area of a circle and represent its circumference. Which of the following equations expresses in terms of ?
Let
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The formula for the area of a circle is , and the formula for circumference is . If we solve for C in terms of r, we get
.
We can then substitute this value of r into the formula for the area:
The formula for the area of a circle is
We can then substitute this value of r into the formula for the area:
The circle above has a radius of , and the measure of is . What is the length of chord ?
The circle above has a radius of
Tap to see back →
To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the side of the model triangle.
Therefore,
Because x is equal to half of the chord, the answer is .
To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle.
Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:
So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the
Therefore,
Because x is equal to half of the chord, the answer is
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?
Tap to see back →
Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Set the area of the circle equal to four times the circumference πr_2 = 4(2_πr).
Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
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For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
The perimeter of a circle is 36 π. What is the diameter of the circle?
The perimeter of a circle is 36 π. What is the diameter of the circle?
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The perimeter of a circle = 2 πr = πd
Therefore d = 36
The perimeter of a circle = 2 πr = πd
Therefore d = 36