Quadrilaterals - Math
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Find the area of a square with a diagonal of 
.
Find the area of a square with a diagonal of
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A few facts need to be known to solve this problem. Observe that the diagonal of the square cuts it into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: 
.
Rearrange an solve for 
.
Now, solve for the area using the formula 
.
A few facts need to be known to solve this problem. Observe that the diagonal of the square cuts it into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse:
Rearrange an solve for
Now, solve for the area using the formula
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If the ratio of the sides of two squares is 
, what is the ratio of the areas of those two squares?
If the ratio of the sides of two squares is
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Express the ratio of the two sides of the squares as 
. The area of each square is one side multiplied by itself, so the ratios of the areas would be 
. The right side of this equation simplifies to a ratio of 
.
Express the ratio of the two sides of the squares as
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What is the area of a square with a diagonal of 
?
What is the area of a square with a diagonal of
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The formula for the area of a square is 
. However, the problem gives us a diagonal and not a side.
Remember that all sides of a square are equal, so the diagonal cuts the square into two equal triangles, each a 
right triangle.
If we use the Pythagorean Theorem, we see:
Plug in our given diagonal to solve.
From here we can plug our answer back into our original equation:
The formula for the area of a square is
Remember that all sides of a square are equal, so the diagonal cuts the square into two equal triangles, each a
If we use the Pythagorean Theorem, we see:
Plug in our given diagonal to solve.
From here we can plug our answer back into our original equation:
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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?
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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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What is the length of a diagonal of a square with a side length 
? Round to the nearest tenth.
What is the length of a diagonal of a square with a side length
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A square is comprised of two 45-45-90 right triangles. The hypotenuse of a 45-45-90 right triangle follows the rule below, where 
is the length of the sides.
In this instance, 
is equal to 6.
A square is comprised of two 45-45-90 right triangles. The hypotenuse of a 45-45-90 right triangle follows the rule below, where
In this instance,
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A square has an area of 36. If all sides are doubled in value, what is the new area?
A square has an area of 36. If all sides are doubled in value, what is the new area?
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Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.
Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.
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What is the length of the diagonal of a 7-by-7 square? (Round to the nearest tenth.)
What is the length of the diagonal of a 7-by-7 square? (Round to the nearest tenth.)
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To find the diagonal of a square we must use the side lengths to create a 90 degree triangle with side lengths of 7 and a hypotenuse which is equal to the diagonal.
We can use the Pythagorean Theorem here to solve for the hypotenuse of a right triangle.
The Pythagorean Theorem states 
, where a and b are the sidelengths and c is the hypotenuse.
Plug the side lengths into the equation as 
and 
:
Square the numbers:
Add the terms on the left side of the equation together:
Take the square root of both sides:
Therefore the length of the diagonal is 9.9.
To find the diagonal of a square we must use the side lengths to create a 90 degree triangle with side lengths of 7 and a hypotenuse which is equal to the diagonal.
We can use the Pythagorean Theorem here to solve for the hypotenuse of a right triangle.
The Pythagorean Theorem states
Plug the side lengths into the equation as
Square the numbers:
Add the terms on the left side of the equation together:
Take the square root of both sides:
Therefore the length of the diagonal is 9.9.
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What is the area of a kite with diagonals of 5 and 7?
What is the area of a kite with diagonals of 5 and 7?
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To find the area of a kite using diagonals you use the following equation 
That diagonals (
and 
)are the lines created by connecting the two sides opposite of each other.
Plug in the diagonals for 
and 
to get 
Then multiply and divide to get the area. 
The answer is 
To find the area of a kite using diagonals you use the following equation
That diagonals (
Plug in the diagonals for
Then multiply and divide to get the area.
The answer is
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Find the area of the following kite:
Find the area of the following kite:
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The formula for the area of a kite is:
Where 
is the length of one diagonal and 
is the length of the other diagonal
Plugging in our values, we get:
The formula for the area of a kite is:
Where
Plugging in our values, we get:
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Find the area of the following kite:
Find the area of the following kite:
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The formula for the area of a kite is:
where 
is the length of one diagonal and 
is the length of another diagonal.
Use the formulas for a 
triangle and a 
triangle to find the lengths of the diagonals. The formula for a 
triangle is 
and the formula for a 
triangle is 
.
Our 
triangle is: 
Our 
triangle is: 
Plugging in our values, we get:
The formula for the area of a kite is:
where
Use the formulas for a
Our
Our
Plugging in our values, we get:
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Find the perimeter of the following kite:
Find the perimeter of the following kite:
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In order to find the length of the two shorter edges, use a Pythagorean triple:
In order to find the length of the two longer edges, use the Pythagorean theorem:
The formula of the perimeter of a kite is:
Plugging in our values, we get:
In order to find the length of the two shorter edges, use a Pythagorean triple:
In order to find the length of the two longer edges, use the Pythagorean theorem:
The formula of the perimeter of a kite is:
Plugging in our values, we get:
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Find the perimeter of the following kite:
Find the perimeter of the following kite:
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The formula for the perimeter of a kite is:
Where 
is the length of the longer side and 
is the length of the shorter side
Use the formulas for a triangle and a triangle to find the lengths of the longer sides. The formula for a triangle is and the formula for a triangle is 
.
Our triangle is:
Our triangle is: 
Plugging in our values, we get:
The formula for the perimeter of a kite is:
Where
Use the formulas for a
Our
Our
Plugging in our values, we get:
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Find the measure of angle 
in the isosceles trapezoid pictured below.
Find the measure of angle
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The sum of the angles in any quadrilateral is 360**°, and the properties of an isosceles trapezoid dictate that the sets of angles adjoined by parallel lines (in this case, the bottom set and top set of angles) are equal. Subtracting 2(72°) from 360°** gives the sum of the two top angles, and dividing the resulting 216**°** by 2 yields the measurement of x, which is 108**°**.
The sum of the angles in any quadrilateral is 360**°, and the properties of an isosceles trapezoid dictate that the sets of angles adjoined by parallel lines (in this case, the bottom set and top set of angles) are equal. Subtracting 2(72°) from 360°** gives the sum of the two top angles, and dividing the resulting 216**°** by 2 yields the measurement of x, which is 108**°**.
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What is the area of this regular trapezoid?
What is the area of this regular trapezoid?
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To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20. The base of the triangle is 3 and the height of the triangle is 4. The area of one triangle is 6. Hence the total area will be 20+6+6=32. If you forget to split the shape into a rectangle and TWO triangles, or if you add the dimensions of the trapezoid, you could arrive at 26 as your answer.
To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20. The base of the triangle is 3 and the height of the triangle is 4. The area of one triangle is 6. Hence the total area will be 20+6+6=32. If you forget to split the shape into a rectangle and TWO triangles, or if you add the dimensions of the trapezoid, you could arrive at 26 as your answer.
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What is the area of the trapezoid above if a = 2, b = 6, and h = 4?
What is the area of the trapezoid above if a = 2, b = 6, and h = 4?
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Area of a Trapezoid = ½(a+b)*h
= ½ (2+6) * 4
= ½ (8) * 4
= 4 * 4 = 16
Area of a Trapezoid = ½(a+b)*h
= ½ (2+6) * 4
= ½ (8) * 4
= 4 * 4 = 16
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A trapezoid has a base of length 4, another base of length s, and a height of length s. A square has sides of length s. What is the value of s such that the area of the trapezoid and the area of the square are equal?
A trapezoid has a base of length 4, another base of length s, and a height of length s. A square has sides of length s. What is the value of s such that the area of the trapezoid and the area of the square are equal?
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In general, the formula for the area of a trapezoid is (1/2)(a + b)(h), where a and b are the lengths of the bases, and h is the length of the height. Thus, we can write the area for the trapezoid given in the problem as follows:
area of trapezoid = (1/2)(4 + s)(s)
Similarly, the area of a square with sides of length a is given by _a_2. Thus, the area of the square given in the problem is _s_2.
We now can set the area of the trapezoid equal to the area of the square and solve for s.
(1/2)(4 + s)(s) = _s_2
Multiply both sides by 2 to eliminate the 1/2.
(4 + s)(s) = 2_s_2
Distribute the s on the left.
4_s_ + _s_2 = 2_s_2
Subtract _s_2 from both sides.
4_s_ = _s_2
Because s must be a positive number, we can divide both sides by s.
4 = s
This means the value of s must be 4.
The answer is 4.
In general, the formula for the area of a trapezoid is (1/2)(a + b)(h), where a and b are the lengths of the bases, and h is the length of the height. Thus, we can write the area for the trapezoid given in the problem as follows:
area of trapezoid = (1/2)(4 + s)(s)
Similarly, the area of a square with sides of length a is given by _a_2. Thus, the area of the square given in the problem is _s_2.
We now can set the area of the trapezoid equal to the area of the square and solve for s.
(1/2)(4 + s)(s) = _s_2
Multiply both sides by 2 to eliminate the 1/2.
(4 + s)(s) = 2_s_2
Distribute the s on the left.
4_s_ + _s_2 = 2_s_2
Subtract _s_2 from both sides.
4_s_ = _s_2
Because s must be a positive number, we can divide both sides by s.
4 = s
This means the value of s must be 4.
The answer is 4.
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This figure is an isosceles trapezoid with bases of 6 in and 18 in and a side of 10 in.
What is the area of the isoceles trapezoid?
This figure is an isosceles trapezoid with bases of 6 in and 18 in and a side of 10 in.
What is the area of the isoceles trapezoid?
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In order to find the area of an isoceles trapezoid, you must average the bases and multiply by the height.
The average of the bases is straight forward:
In order to find the height, you must draw an altitude. This creates a right triangle in which one of the legs is also the height of the trapezoid. You may recognize the Pythagorean triple (6-8-10) and easily identify the height as 8. Otherwise, use 
.
Multiply the average of the bases (12) by the height (8) to get an area of 96.
In order to find the area of an isoceles trapezoid, you must average the bases and multiply by the height.
The average of the bases is straight forward:
In order to find the height, you must draw an altitude. This creates a right triangle in which one of the legs is also the height of the trapezoid. You may recognize the Pythagorean triple (6-8-10) and easily identify the height as 8. Otherwise, use
Multiply the average of the bases (12) by the height (8) to get an area of 96.
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Find the area of the following trapezoid:
Find the area of the following trapezoid:
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The formula for the area of a trapezoid is:
Where 
is the length of one base, 
is the length of the other base, and 
is the height.
To find the height of the trapezoid, use a Pythagorean triple:
Plugging in our values, we get:
The formula for the area of a trapezoid is:
Where
To find the height of the trapezoid, use a Pythagorean triple:
Plugging in our values, we get:
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Find the area of the following trapezoid:
Find the area of the following trapezoid:
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Use the formula for 
triangles in order to find the length of the bottom base and the height.
The formula is:
Where 
is the length of the side opposite the 
.
Beginning with the 
side, if we were to create a 
triangle, the length of the base is 
, and the height is 
.
Creating another triangle on the left, we find the height is , the length of the base is 
, and the side is 
.
The formula for the area of a trapezoid is:
Where is the length of one base, is the length of the other base, and is the height.
Plugging in our values, we get:
Use the formula for
The formula is:
Where
Beginning with the
Creating another
The formula for the area of a trapezoid is:
Where
Plugging in our values, we get:
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Determine the area of the following trapezoid:
Determine the area of the following trapezoid:
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The formula for the area of a trapezoid is:
,
where 
is the length of one base, 
is the length of another base, and 
is the length of the height.
Plugging in our values, we get:
The formula for the area of a trapezoid is:
where
Plugging in our values, we get:
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