DSQ: Calculating the area of a polygon - GMAT Quantitative

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

and multiplying the result by 6.

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.

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 .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by 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 . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

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 , we can calculate the radius from the area formula (divide by , 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 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 is known to be 8, the length of each leg can be found using the formula

Since is a 45-45-90 triangle, multiply this leg length by to get , the length of one side; this is sufficient to get the area of the octagon.

S= $\frac{1}{2}$times L1times L2 (L1 and 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.

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

and multiplying the result by 6.

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.

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 .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by 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 . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

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 , we can calculate the radius from the area formula (divide by , 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 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 is known to be 8, the length of each leg can be found using the formula

Since is a 45-45-90 triangle, multiply this leg length by to get , the length of one side; this is sufficient to get the area of the octagon.

S= $\frac{1}{2}$times L1times L2 (L1 and 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.

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

and multiplying the result by 6.

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.

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 .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by 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 . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

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 , we can calculate the radius from the area formula (divide by , 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 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 is known to be 8, the length of each leg can be found using the formula

Since is a 45-45-90 triangle, multiply this leg length by to get , the length of one side; this is sufficient to get the area of the octagon.

S= $\frac{1}{2}$times L1times L2 (L1 and 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.

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

and multiplying the result by 6.

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.

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 .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by 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 . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

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 , we can calculate the radius from the area formula (divide by , 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 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 is known to be 8, the length of each leg can be found using the formula

Since is a 45-45-90 triangle, multiply this leg length by to get , the length of one side; this is sufficient to get the area of the octagon.