Polygons - ACT Math
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What is the number of sides of a regular polygon with each exterior angle $30$?
What is the number of sides of a regular polygon with each exterior angle $30$?
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$12$. Since $\frac{360°}{30°} = 12$ sides for the polygon.
$12$. Since $\frac{360°}{30°} = 12$ sides for the polygon.
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Calculate the area of a rhombus with diagonals 10 and 8.
Calculate the area of a rhombus with diagonals 10 and 8.
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- Apply $\frac{10 \times 8}{2} = \frac{80}{2} = 40$.
- Apply $\frac{10 \times 8}{2} = \frac{80}{2} = 40$.
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What is the number of sides of a regular polygon with each exterior angle $24$?
What is the number of sides of a regular polygon with each exterior angle $24$?
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$15$. Since $\frac{360°}{24°} = 15$ sides for the polygon.
$15$. Since $\frac{360°}{24°} = 15$ sides for the polygon.
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What is the measure of each exterior angle of a regular $n$-gon?
What is the measure of each exterior angle of a regular $n$-gon?
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$\frac{360}{n}$. Total exterior angle sum divided by number of vertices.
$\frac{360}{n}$. Total exterior angle sum divided by number of vertices.
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What is each interior angle of a regular polygon with $n=20$ sides?
What is each interior angle of a regular polygon with $n=20$ sides?
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$162$. Using $\frac{180(20-2)}{20} = \frac{3240}{20} = 162°$.
$162$. Using $\frac{180(20-2)}{20} = \frac{3240}{20} = 162°$.
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What is each exterior angle of a regular polygon with $n=18$ sides?
What is each exterior angle of a regular polygon with $n=18$ sides?
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$20$. Using $\frac{360°}{18} = 20°$ for each exterior angle.
$20$. Using $\frac{360°}{18} = 20°$ for each exterior angle.
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What is the measure of an exterior angle if the interior angle is $112$?
What is the measure of an exterior angle if the interior angle is $112$?
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$68$. Interior and exterior angles are supplementary: $180° - 112° = 68°$.
$68$. Interior and exterior angles are supplementary: $180° - 112° = 68°$.
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What is the interior angle if the exterior angle of a convex polygon is $55$?
What is the interior angle if the exterior angle of a convex polygon is $55$?
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$125$. Interior and exterior angles are supplementary: $180° - 55° = 125°$.
$125$. Interior and exterior angles are supplementary: $180° - 55° = 125°$.
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What is the sum of interior angles of a polygon with $15$ sides?
What is the sum of interior angles of a polygon with $15$ sides?
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$2340$. Using $180(15-2) = 180(13) = 2340°$.
$2340$. Using $180(15-2) = 180(13) = 2340°$.
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What is a four-sided polygon called?
What is a four-sided polygon called?
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Quadrilateral. The prefix 'quad-' means four in Latin.
Quadrilateral. The prefix 'quad-' means four in Latin.
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What is the definition of a convex polygon?
What is the definition of a convex polygon?
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All interior angles are less than $180$. No interior angle exceeds a straight angle.
All interior angles are less than $180$. No interior angle exceeds a straight angle.
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What is the definition of a diagonal of a polygon?
What is the definition of a diagonal of a polygon?
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A segment joining two nonadjacent vertices. Connects vertices that don't share a side.
A segment joining two nonadjacent vertices. Connects vertices that don't share a side.
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What is the sum of the interior angles of a hexagon?
What is the sum of the interior angles of a hexagon?
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720°. Apply $(6-2) \times 180° = 4 \times 180° = 720°$.
720°. Apply $(6-2) \times 180° = 4 \times 180° = 720°$.
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What is the definition of a concave polygon?
What is the definition of a concave polygon?
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At least one interior angle is greater than $180$. Contains at least one reflex interior angle.
At least one interior angle is greater than $180$. Contains at least one reflex interior angle.
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What is the area of a regular polygon with apothem $10$ and perimeter $36$?
What is the area of a regular polygon with apothem $10$ and perimeter $36$?
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$180$. Using $\frac{1}{2} \cdot 10 \cdot 36 = \frac{360}{2} = 180$.
$180$. Using $\frac{1}{2} \cdot 10 \cdot 36 = \frac{360}{2} = 180$.
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What is the number of triangles formed by drawing diagonals from one vertex of a convex $n$-gon?
What is the number of triangles formed by drawing diagonals from one vertex of a convex $n$-gon?
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$n-2$. Diagonals from one vertex create $(n-2)$ triangles.
$n-2$. Diagonals from one vertex create $(n-2)$ triangles.
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What is the apothem $a$ of a regular polygon with area $150$ and perimeter $50$?
What is the apothem $a$ of a regular polygon with area $150$ and perimeter $50$?
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$6$. From $150 = \frac{1}{2} \cdot a \cdot 50$, solving gives $a = 6$.
$6$. From $150 = \frac{1}{2} \cdot a \cdot 50$, solving gives $a = 6$.
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What is the number of sides of a regular polygon with each exterior angle $24$?
What is the number of sides of a regular polygon with each exterior angle $24$?
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$15$. Since $\frac{360°}{24°} = 15$ sides for the polygon.
$15$. Since $\frac{360°}{24°} = 15$ sides for the polygon.
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What is the number of diagonals in a decagon ($n=10$)?
What is the number of diagonals in a decagon ($n=10$)?
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$35$. Using $\frac{10(10-3)}{2} = \frac{10 \cdot 7}{2} = 35$.
$35$. Using $\frac{10(10-3)}{2} = \frac{10 \cdot 7}{2} = 35$.
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What is the number of sides $n$ if the sum of interior angles is $1260$?
What is the number of sides $n$ if the sum of interior angles is $1260$?
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$9$. From $180(n-2) = 1260$, solving gives $n = 9$.
$9$. From $180(n-2) = 1260$, solving gives $n = 9$.
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What is each interior angle of a regular octagon?
What is each interior angle of a regular octagon?
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$135$. Using $\frac{180(8-2)}{8} = \frac{1080}{8} = 135°$.
$135$. Using $\frac{180(8-2)}{8} = \frac{1080}{8} = 135°$.
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What is the sum of interior angles of an octagon?
What is the sum of interior angles of an octagon?
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$1080$. Using $180(8-2) = 180(6) = 1080°$.
$1080$. Using $180(8-2) = 180(6) = 1080°$.
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What is the sum of interior angles of a dodecagon ($n=12$)?
What is the sum of interior angles of a dodecagon ($n=12$)?
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$1800$. Using $180(12-2) = 180(10) = 1800°$.
$1800$. Using $180(12-2) = 180(10) = 1800°$.
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What is the perimeter $P$ of a regular polygon with area $96$ and apothem $8$?
What is the perimeter $P$ of a regular polygon with area $96$ and apothem $8$?
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$24$. From $96 = \frac{1}{2} \cdot 8 \cdot P$, solving gives $P = 24$.
$24$. From $96 = \frac{1}{2} \cdot 8 \cdot P$, solving gives $P = 24$.
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Identify the polygon with all sides and angles congruent.
Identify the polygon with all sides and angles congruent.
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Regular polygon. All sides are congruent and all angles are congruent.
Regular polygon. All sides are congruent and all angles are congruent.
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What is a polygon with three sides called?
What is a polygon with three sides called?
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Triangle. The simplest polygon with three vertices and sides.
Triangle. The simplest polygon with three vertices and sides.
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Identify the shape: A quadrilateral with only one pair of parallel sides.
Identify the shape: A quadrilateral with only one pair of parallel sides.
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Trapezoid. Has exactly one pair of parallel sides called bases.
Trapezoid. Has exactly one pair of parallel sides called bases.
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What is the formula for the area of a rhombus?
What is the formula for the area of a rhombus?
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Area = $\frac{d_1 \times d_2}{2}$. Uses the product of diagonals divided by 2.
Area = $\frac{d_1 \times d_2}{2}$. Uses the product of diagonals divided by 2.
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What is the term for a polygon with four sides and four angles?
What is the term for a polygon with four sides and four angles?
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Quadrilateral. A polygon with exactly four sides and four vertices.
Quadrilateral. A polygon with exactly four sides and four vertices.
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What is the measure of each exterior angle in a regular hexagon?
What is the measure of each exterior angle in a regular hexagon?
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60°. Apply $\frac{360°}{6} = 60°$ for each exterior angle.
60°. Apply $\frac{360°}{6} = 60°$ for each exterior angle.
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