DSQ: Calculating the area of a polygon - GMAT Quantitative

Card 1 of 40

Didn't Know

Knew It

1 of 2019 left

Question

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

Tap to reveal answer

Answer

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ and $\dpi{100} \small L2$ are the diagonals of the rhombus). From statement (1) we cannot get to the lengths of diagonals. From statement (2) we only know the length of one diagonal, which is insufficient. However, putting the two statements together, we can use the Pythagorean Theorem to calculate the other diagonal, and then use the formula to calculate the area.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

Tap to reveal answer

Answer

A regular hexagon can be viewed as the composite of six equilateral triangles. The sidelength of each of the triangles is equal to both the sidelength of the hexagon (one-sixth of the perimeter) and the radius of the hexagon. From either statement, it is possible to find the area of one triangle by substituting in the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and multiplying the result by 6.

← Didn't Know|Knew It →

Question

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

Tap to reveal answer

Answer

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Tap to reveal answer

Answer

A regular hexagon can be viewed as a composite of six equilateral triangles, each of whose sidelength is the radius - the distance from its center to a vertex - of the hexagon. If the radius of the hexagon is known, then the area of the hexagon can be calculated to be $A = 6 \cdot \frac{r^{2}\sqrt{3}}{4} = \frac{3r^{2}\sqrt{3}}{2}$ .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by $\pi$ and extracting the square root). This length coincides with the height of each equilateral triangle. From there, the 30-60-90 Theorem can be used to find the sidelength of each triangle, and the area of the hexagon follows.

From Statement 2 alone, the radius of the circumscribed circle, or circumcircle, can be found by dividing its circumference by $2 \pi$ . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

← Didn't Know|Knew It →

Question

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

Tap to reveal answer

Answer

Quadrilateral is a square; each of its diagonals is a diameter of its circumscribed circle, or circumcircle. Therefore, if we know the area of its circumcircle from Statement 1 to be $32 \pi$ , we can calculate the radius from the area formula (divide by $\pi$ , extract the square root of the quotient). Twice this is the diameter, which is also the length of a diagonal of this square. Divide this by $\sqrt{2}$ to get . This is also equal to , the length of one side; this is sufficient to get the area of the octagon.

From Statement 2 alone, since the area of isosceles triangle $\Delta BXC$ is known to be 8, the length of each leg can be found using the formula

$\frac{1}{2} s^{2} = 16$

Since $\Delta BXC$ is a 45-45-90 triangle, multiply this leg length by $\sqrt{2}$ to get , the length of one side; this is sufficient to get the area of the octagon.

← Didn't Know|Knew It →

Question

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

Tap to reveal answer

Answer

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ and $\dpi{100} \small L2$ are the diagonals of the rhombus). From statement (1) we cannot get to the lengths of diagonals. From statement (2) we only know the length of one diagonal, which is insufficient. However, putting the two statements together, we can use the Pythagorean Theorem to calculate the other diagonal, and then use the formula to calculate the area.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

Tap to reveal answer

Answer

A regular hexagon can be viewed as the composite of six equilateral triangles. The sidelength of each of the triangles is equal to both the sidelength of the hexagon (one-sixth of the perimeter) and the radius of the hexagon. From either statement, it is possible to find the area of one triangle by substituting in the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and multiplying the result by 6.

← Didn't Know|Knew It →

Question

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

Tap to reveal answer

Answer

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Tap to reveal answer

Answer

A regular hexagon can be viewed as a composite of six equilateral triangles, each of whose sidelength is the radius - the distance from its center to a vertex - of the hexagon. If the radius of the hexagon is known, then the area of the hexagon can be calculated to be $A = 6 \cdot \frac{r^{2}\sqrt{3}}{4} = \frac{3r^{2}\sqrt{3}}{2}$ .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by $\pi$ and extracting the square root). This length coincides with the height of each equilateral triangle. From there, the 30-60-90 Theorem can be used to find the sidelength of each triangle, and the area of the hexagon follows.

From Statement 2 alone, the radius of the circumscribed circle, or circumcircle, can be found by dividing its circumference by $2 \pi$ . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

← Didn't Know|Knew It →

Question

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

Tap to reveal answer

Answer

Quadrilateral is a square; each of its diagonals is a diameter of its circumscribed circle, or circumcircle. Therefore, if we know the area of its circumcircle from Statement 1 to be $32 \pi$ , we can calculate the radius from the area formula (divide by $\pi$ , extract the square root of the quotient). Twice this is the diameter, which is also the length of a diagonal of this square. Divide this by $\sqrt{2}$ to get . This is also equal to , the length of one side; this is sufficient to get the area of the octagon.

From Statement 2 alone, since the area of isosceles triangle $\Delta BXC$ is known to be 8, the length of each leg can be found using the formula

$\frac{1}{2} s^{2} = 16$

Since $\Delta BXC$ is a 45-45-90 triangle, multiply this leg length by $\sqrt{2}$ to get , the length of one side; this is sufficient to get the area of the octagon.

← Didn't Know|Knew It →

Question

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

Tap to reveal answer

Answer

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ and $\dpi{100} \small L2$ are the diagonals of the rhombus). From statement (1) we cannot get to the lengths of diagonals. From statement (2) we only know the length of one diagonal, which is insufficient. However, putting the two statements together, we can use the Pythagorean Theorem to calculate the other diagonal, and then use the formula to calculate the area.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

Tap to reveal answer

Answer

A regular hexagon can be viewed as the composite of six equilateral triangles. The sidelength of each of the triangles is equal to both the sidelength of the hexagon (one-sixth of the perimeter) and the radius of the hexagon. From either statement, it is possible to find the area of one triangle by substituting in the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and multiplying the result by 6.

← Didn't Know|Knew It →

Question

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

Tap to reveal answer

Answer

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Tap to reveal answer

Answer

A regular hexagon can be viewed as a composite of six equilateral triangles, each of whose sidelength is the radius - the distance from its center to a vertex - of the hexagon. If the radius of the hexagon is known, then the area of the hexagon can be calculated to be $A = 6 \cdot \frac{r^{2}\sqrt{3}}{4} = \frac{3r^{2}\sqrt{3}}{2}$ .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by $\pi$ and extracting the square root). This length coincides with the height of each equilateral triangle. From there, the 30-60-90 Theorem can be used to find the sidelength of each triangle, and the area of the hexagon follows.

From Statement 2 alone, the radius of the circumscribed circle, or circumcircle, can be found by dividing its circumference by $2 \pi$ . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

← Didn't Know|Knew It →

Question

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

Tap to reveal answer

Answer

Quadrilateral is a square; each of its diagonals is a diameter of its circumscribed circle, or circumcircle. Therefore, if we know the area of its circumcircle from Statement 1 to be $32 \pi$ , we can calculate the radius from the area formula (divide by $\pi$ , extract the square root of the quotient). Twice this is the diameter, which is also the length of a diagonal of this square. Divide this by $\sqrt{2}$ to get . This is also equal to , the length of one side; this is sufficient to get the area of the octagon.

From Statement 2 alone, since the area of isosceles triangle $\Delta BXC$ is known to be 8, the length of each leg can be found using the formula

$\frac{1}{2} s^{2} = 16$

Since $\Delta BXC$ is a 45-45-90 triangle, multiply this leg length by $\sqrt{2}$ to get , the length of one side; this is sufficient to get the area of the octagon.

← Didn't Know|Knew It →

Question

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

Tap to reveal answer

Answer

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ and $\dpi{100} \small L2$ are the diagonals of the rhombus). From statement (1) we cannot get to the lengths of diagonals. From statement (2) we only know the length of one diagonal, which is insufficient. However, putting the two statements together, we can use the Pythagorean Theorem to calculate the other diagonal, and then use the formula to calculate the area.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

Tap to reveal answer

Answer

A regular hexagon can be viewed as the composite of six equilateral triangles. The sidelength of each of the triangles is equal to both the sidelength of the hexagon (one-sixth of the perimeter) and the radius of the hexagon. From either statement, it is possible to find the area of one triangle by substituting in the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and multiplying the result by 6.

← Didn't Know|Knew It →

Question

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

Tap to reveal answer

Answer

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

← Didn't Know|Knew It →

Question

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Tap to reveal answer

Answer

A regular hexagon can be viewed as a composite of six equilateral triangles, each of whose sidelength is the radius - the distance from its center to a vertex - of the hexagon. If the radius of the hexagon is known, then the area of the hexagon can be calculated to be $A = 6 \cdot \frac{r^{2}\sqrt{3}}{4} = \frac{3r^{2}\sqrt{3}}{2}$ .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by $\pi$ and extracting the square root). This length coincides with the height of each equilateral triangle. From there, the 30-60-90 Theorem can be used to find the sidelength of each triangle, and the area of the hexagon follows.

From Statement 2 alone, the radius of the circumscribed circle, or circumcircle, can be found by dividing its circumference by $2 \pi$ . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

← Didn't Know|Knew It →

Question

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

Tap to reveal answer

Answer

Quadrilateral is a square; each of its diagonals is a diameter of its circumscribed circle, or circumcircle. Therefore, if we know the area of its circumcircle from Statement 1 to be $32 \pi$ , we can calculate the radius from the area formula (divide by $\pi$ , extract the square root of the quotient). Twice this is the diameter, which is also the length of a diagonal of this square. Divide this by $\sqrt{2}$ to get . This is also equal to , the length of one side; this is sufficient to get the area of the octagon.

From Statement 2 alone, since the area of isosceles triangle $\Delta BXC$ is known to be 8, the length of each leg can be found using the formula

$\frac{1}{2} s^{2} = 16$

Since $\Delta BXC$ is a 45-45-90 triangle, multiply this leg length by $\sqrt{2}$ to get , the length of one side; this is sufficient to get the area of the octagon.

← Didn't Know|Knew It →

Knew It

DSQ: Calculating the area of a polygon - GMAT Quantitative

What is the area of a parallelogram with four equal sides? (1) Each side is (2) One diagonal is

What is the area of a regular hexagon? Statement 1: The perimeter of the hexagon is 48. Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid? Statement 1: The length of its midsegment is 20. Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula: Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon? Statement 1: The area of the circle inscribed inside the hexagon is . Statement 2: The circumference of the circle that is circumscribed about the hexagon is .

Give the area of the above regular octagon. Statement 1: The circle that circumscribes Quadrilateral has area . Statement 2: has area 16.

What is the area of a parallelogram with four equal sides? (1) Each side is (2) One diagonal is

What is the area of a regular hexagon? Statement 1: The perimeter of the hexagon is 48. Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid? Statement 1: The length of its midsegment is 20. Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula: Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon? Statement 1: The area of the circle inscribed inside the hexagon is . Statement 2: The circumference of the circle that is circumscribed about the hexagon is .

Give the area of the above regular octagon. Statement 1: The circle that circumscribes Quadrilateral has area . Statement 2: has area 16.

What is the area of a parallelogram with four equal sides? (1) Each side is (2) One diagonal is

What is the area of a regular hexagon? Statement 1: The perimeter of the hexagon is 48. Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid? Statement 1: The length of its midsegment is 20. Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula: Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon? Statement 1: The area of the circle inscribed inside the hexagon is . Statement 2: The circumference of the circle that is circumscribed about the hexagon is .

Give the area of the above regular octagon. Statement 1: The circle that circumscribes Quadrilateral has area . Statement 2: has area 16.

What is the area of a parallelogram with four equal sides? (1) Each side is (2) One diagonal is

What is the area of a regular hexagon? Statement 1: The perimeter of the hexagon is 48. Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid? Statement 1: The length of its midsegment is 20. Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula: Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon? Statement 1: The area of the circle inscribed inside the hexagon is . Statement 2: The circumference of the circle that is circumscribed about the hexagon is .

Give the area of the above regular octagon. Statement 1: The circle that circumscribes Quadrilateral has area . Statement 2: has area 16.

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.