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Quizzes > Mathematics > Geometry

Ultimate Polygons & Quadrilaterals Quiz - Test Your Knowledge

Think you can ace this Unit 8 test on polygons and quadrilaterals? Let's begin!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper cut art polygons quadrilaterals arranged on dark blue background promoting free scored quiz

This test on quadrilaterals helps you practice four‑sided shapes and get exam‑ready. You get instant scoring and clear feedback as you review parallel sides, angle sums, and diagonals, so you can spot gaps before the test. For extra help, see the study guide or try the full quiz .

What is the sum of the interior angles of any quadrilateral?
540°
360°
720°
180°
The sum of interior angles of a quadrilateral is given by (n?2)×180° = (4?2)×180° = 360°. This holds for all simple quadrilaterals, whether convex or concave. You can derive this by dividing the quadrilateral into two triangles. See more at .
Which quadrilateral has four equal sides and four right angles?
Square
Parallelogram
Rhombus
Rectangle
A square has all four sides equal and all four interior angles equal to 90°. A rhombus has equal sides but not necessarily right angles, and a rectangle has right angles but not necessarily equal sides. This makes the square the only quadrilateral meeting both criteria. More details at .
In a parallelogram, opposite sides are always:
Parallel and equal in length
Perpendicular
Parallel only
Equal in length only
By definition, a parallelogram has two pairs of parallel sides, and it also follows that opposite sides are congruent (equal in length). This is a key property that distinguishes parallelograms from other quadrilaterals. See for proof and properties.
Which quadrilateral has exactly one pair of parallel sides?
Trapezoid (US) / Trapezium (UK)
Kite
Rectangle
Parallelogram
A trapezoid (US) or trapezium (UK) is defined as having exactly one pair of parallel sides. Parallelograms have two pairs of parallel sides, while kites and rectangles have different defining properties. For more, visit .
Which quadrilateral has two pairs of adjacent equal sides but no parallel sides?
Isosceles trapezoid
Kite
Rhombus
Square
A kite has two distinct pairs of adjacent sides that are equal, and its sides are not parallel. A rhombus has four equal sides (two pairs of adjacent equal sides) but also parallel opposite sides, so it does not meet the criterion. See for more.
What is the sum of the exterior angles (one at each vertex) of any convex quadrilateral?
360°
720°
180°
90°
The sum of the exterior angles of any convex polygon, taken one per vertex, is always 360°, regardless of the number of sides. This follows from the interior angles summing to (n?2)×180° and the linear pair with each exterior angle. See .
In a rectangle, the diagonals are always:
Equal in length only
Equal in length and bisect each other
Bisect angles
Perpendicular
In a rectangle, each diagonal has the same length (by symmetry) and they bisect each other (each diagonal cuts the other into two equal parts). They do not necessarily bisect the angles or meet at right angles. More at .
Which statement about the diagonals of a parallelogram is always true?
They bisect the interior angles
They are perpendicular
They bisect each other
They are equal in length
In any parallelogram, the diagonals bisect each other, meaning each diagonal cuts the other into two equal segments. This follows from the parallelism and congruent triangles in the figure. They do not necessarily bisect angles, nor are they always perpendicular or equal. See .
Which quadrilateral has perpendicular diagonals that also bisect its interior angles, but the diagonals are not necessarily equal in length?
Kite
Rectangle
Square
Rhombus
A rhombus has all sides equal, its diagonals intersect at right angles, and each diagonal bisects the vertex angles. While a square shares these properties, its diagonals are also equal, making the rhombus the only choice fitting exactly the description. See .
Which of the following quadrilaterals is always cyclic (all vertices lie on a single circle)?
Kite
Rhombus
Rectangle
General parallelogram
All rectangles are cyclic because each pair of opposite angles is supplementary (sum to 180°), which is the criterion for a quadrilateral to be cyclic. Rhombi and general parallelograms do not in general have opposite angles summing to 180°. More at .
What is the area of a parallelogram with base b and height h?
b × h
½ b × h
b + h
2 b × h
The area of a parallelogram is given by the product of its base and height (perpendicular distance between bases). This is because you can decompose it into a rectangle of the same base and height. See .
Given quadrilateral ABCD with vertices A(0,0), B(4,0), C(5,3) and D(1,3), what type of quadrilateral is ABCD?
Parallelogram
Rhombus
Trapezoid
Rectangle
Vectors AB=(4,0) and CD=(?4,0) are parallel and equal, and BC=(1,3) and AD=(1,3) are also parallel and equal, so opposite sides are both parallel and congruent, which defines a parallelogram. It is not a rectangle because angles are not all right. See .
If three interior angles of a quadrilateral measure 90°, 120°, and 80°, what is the measure of the fourth angle?
80°
60°
75°
70°
The sum of interior angles of a quadrilateral is 360°. Subtracting the sum of the three given angles (90+120+80=290) from 360 gives 70°. Hence the fourth angle is 70°. More at .
Which of the following is always true for a kite?
It has two pairs of adjacent equal sides
Its opposite sides are equal
All its angles are right angles
Its diagonals are equal
By definition, a kite has two distinct pairs of adjacent sides that are congruent. Its opposite sides are not necessarily equal, and its diagonals are not both equal in general, though they are perpendicular. See .
In a trapezoid with bases of lengths b? and b?, the length of the mid-segment (the segment connecting the midpoints of the legs) is:
2(b? + b?)
(b? + b?) / 2
b? + b?
|b? ? b?| / 2
The mid-segment of a trapezoid is parallel to the bases and its length equals the average of the lengths of the two bases: (b? + b?)/2. This follows from parallel line segment properties. See .
According to Bretschneider's formula, the area K of any quadrilateral with sides a, b, c, d and opposite angles A and C is:
?((s?a)(s?b)(s?c)(s?d) ? abcd·cos²((A+C)/2))
abcd·sin((A+C)/2)
s(s?a)(s?b)(s?c)(s?d)
(a+b+c+d)/2
Bretschneider's formula generalizes Brahmagupta's formula to non-cyclic quadrilaterals: K = ?((s?a)(s?b)(s?c)(s?d) ? abcd·cos²((A+C)/2)), where s is the semiperimeter. The extra cosine term accounts for the deviation from cyclicity. See .
Pitot's theorem states that a quadrilateral has an incircle if and only if:
The sums of lengths of opposite sides are equal
All sides are equal
Opposite angles are equal
Diagonals are perpendicular
Pitot's theorem for tangential quadrilaterals (those with an incircle) states that the sum of the lengths of one pair of opposite sides equals the sum of the lengths of the other pair. This is both necessary and sufficient. More at .
In a cyclic quadrilateral, the sum of each pair of opposite angles is always:
360°
180°
Equal to each other
90°
A quadrilateral is cyclic if and only if each pair of opposite angles sums to 180°. This is the defining property of cyclic quadrilaterals inscribed in a circle. See .
In a rhombus with one interior angle measuring 60°, what is the ratio of the lengths of its diagonals?
1 : 2
1 : ?3
?3 : 1
2 : 1
For a rhombus of side s and angle A=60°, the diagonals are d? = s?(2?2cosA)=s?1=s and d? = s?(2+2cosA)=s?3. Hence the ratio is d?:d? = 1:?3. See .
Using the shoelace formula, what is the area of the quadrilateral with vertices (0,0), (4,0), (4,3), (0,3)?
12
7.5
10
14
The given points form a 4×3 rectangle, so its area is 4×3 = 12. Using the shoelace formula on the coordinates yields the same result. See .
Which of the following is true for the quadrilateral with vertices A(0,0), B(2,2), C(4,0), D(2,?2)?
It is a square
It is a parallelogram
It is cyclic
It is a rectangle
All four vertices lie at the same distance (2?2) from the point (2,0), so they lie on a common circle and the quadrilateral is cyclic. It is not a rectangle or square because only opposite angles sum to 180°, not each angle being 90°. See .
According to Varignon's theorem, the quadrilateral formed by joining the midpoints of the sides of any quadrilateral is always a:
Rectangle
Parallelogram
Rhombus
Square
Varignon's theorem states that connecting the midpoints of the sides of any quadrilateral in order yields a parallelogram. This is true whether or not the original quadrilateral is convex. Learn more at .
A quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if one of the following holds. Which one?
Sum of two adjacent angles is 180°
Sum of the squares of two opposite sides equals the sum of the squares of the other two
Sum of lengths of one pair of opposite sides equals the other pair
Diagonals are equal in length
A quadrilateral is orthodiagonal if and only if a² + c² = b² + d², where a, b, c, d are the side lengths in order. This condition ensures the diagonals meet at right angles. See .
Which formula gives the area of a cyclic quadrilateral with side lengths a, b, c, d and semiperimeter s?
½(a+c)(b+d)
?((s?a)(s?b)(s?c)(s?d) ? abcd·cos²((A+C)/2))
?((s?a)(s?b)(s?c)(s?d))
ab + cd
Brahmagupta's formula for the area of a cyclic quadrilateral simplifies Bretschneider's formula by removing the cosine term: K = ?((s?a)(s?b)(s?c)(s?d)). This applies only when the quadrilateral is cyclic. More at .
Under what condition does Bretschneider's formula reduce to Brahmagupta's formula for a quadrilateral's area?
When all four sides are equal
When diagonals are perpendicular
When the quadrilateral is convex
When the sum of two opposite angles equals 180°
Bretschneider's formula includes a term with cos²((A+C)/2). It reduces to Brahmagupta's formula exactly when A+C=180°, i.e., the quadrilateral is cyclic. In that case, cos((A+C)/2)=cos(90°)=0, removing the extra term. See .
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Study Outcomes

  1. Identify Quadrilateral Types -

    Distinguish between parallelograms, rectangles, squares, rhombi, and trapezoids by recognizing defining side lengths and angle properties.

  2. Classify Polygons and Quadrilaterals -

    Use key attributes such as parallel sides and angle measures to categorize shapes for your unit 8 test on polygons and quadrilaterals.

  3. Apply Angle-Sum Properties -

    Calculate interior and exterior angles of quadrilaterals using the angle-sum theorem to solve geometry problems accurately.

  4. Calculate Perimeter and Area -

    Compute lengths, perimeters, and areas of various quadrilaterals by applying appropriate formulas and units.

  5. Analyze Test on Quadrilaterals Results -

    Interpret your quiz scores and use polygons and quadrilaterals test answers to pinpoint strengths and target areas for improvement.

  6. Develop Test-Taking Strategies -

    Build effective approaches and time-management techniques to maximize your performance on a test on quadrilaterals and related practice quizzes.

Cheat Sheet

  1. Classification and Hierarchy -

    Review how quadrilaterals are classified - parallelogram, rectangle, rhombus, square, and trapezoid - by side lengths and angle properties. Use the mnemonic "P-P-R-S-T" (Parallelogram, Rectangle, Square, Trapezoid) to remember that each inherits features from the previous shape. This clear hierarchy is essential for acing your unit 8 test on polygons and quadrilaterals.

  2. Parallelogram Essentials -

    Memorize that in any parallelogram opposite sides are congruent and opposite angles are equal, and the diagonals bisect each other. A quick coordinate check: if A(0,0), B(b,0), D(d,e), then C = (b+d, e) ensures midpoints of AC and BD coincide. These core properties form the backbone of many problems on a test on quadrilaterals.

  3. Area Formulas -

    Know A = base × height for parallelograms, A = ½ × (d₝ × d₂) for kites, and A = ½ × (b₝ + b₂) × height for trapezoids. For example, a trapezoid with bases 8 and 12 and height 5 has area A = ½×(8+12)×5 = 50. These formulas will make your quadrilaterals practice quiz a breeze.

  4. Angle-Sum and Exterior Angles -

    Recall that the sum of interior angles of any quadrilateral is 360°, and the sum of the exterior angles always equals 360° when taken one at each vertex. Use this fact to quickly find missing angles - if three are known, subtract their sum from 360°. When reviewing polygons and quadrilaterals test answers from university geometry guides, this shortcut saves valuable time.

  5. Diagonal Behaviors -

    Differentiate diagonal properties: in rectangles diagonals are equal, in rhombi they're perpendicular, and in squares both rules apply. To remember this, think "RECT angles for equal, RHOMB rules for right (90°)". Practicing these in every quadrilaterals practice quiz builds the intuition you need to solve complex diagonal problems on test day.

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