30 Geometry Questions: No Protractor Needed

[C] Trapezoid | Think of it as the rebellious cousin of the parallelogram family. While parallelograms insist on two parallel pairs, trapezoids keep things interesting with just one, making them perfect for architectural designs and frustrating geometry students everywhere.

[B] 720° | Hexagons are nature's favorite shape (just ask the bees!). To find interior angle sums, use the formula (n-2) × 180°, where n is the number of sides. With 6 sides, you get 4 × 180° = 720°.

[B] 6 | When y-coordinates match, you're dealing with a horizontal line. Simply subtract the x-coordinates: 4 - (-2) = 6. No Pythagorean theorem needed when you're traveling in straight lines across the grid!

[D] Reflection | Reflections are geometry's mirror selfies. They flip everything backwards, which is why your reflection can't high-five you properly. Translations slide, rotations spin, and dilations zoom, but only reflections reverse the world.

[C] 10π | Remember this golden rule: circumference equals π times diameter (C = πd). With diameter 10, you get 10π. It's like measuring the distance around a perfectly round cookie using mathematical ribbon!

[A] 45π | Picture stacking circular pancakes! Volume equals πr²h, so π × 9 × 5 = 45π. That's enough mathematical space to store approximately zero actual pancakes because math doesn't care about your breakfast needs.

[A] 108° | Pentagons are the five-sided stars of geometry. Divide the total interior angles (540°) by 5 vertices, and voilà! Each angle measures 108°, making pentagons slightly more obtuse than your average square.

[B] Corresponding angles are equal | Similar triangles are like family members at different ages. They maintain the same angular features (personality traits) even though their sizes differ. Areas and perimeters scale differently, but angles stay constant.

[B] Isosceles | With two matching sides of 7, this triangle plays favorites. Isosceles triangles are the geometry equivalent of twins with a unique sibling, creating symmetry without going full equilateral commitment.

[C] 2 | Rise over run gives us (10-2)/(5-1) = 8/4 = 2. This line climbs two units up for every unit across, like a determined mountain goat with excellent mathematical precision.

[B] (1, 1) | Finding midpoints is like splitting the check evenly. Average the x-coordinates: (-3+5)/2 = 1, then the y-coordinates: (4-2)/2 = 1. Meeting in the middle never felt so mathematically satisfying!

[D] 360° | No matter how many sides your polygon has, exterior angles always sum to 360°. It's geometry's way of saying every closed journey brings you full circle, literally and mathematically.

[D] 36π | Area equals πr², giving us π × 36. Imagine painting a circular canvas with radius 6. You'd need 36π square units of paint, though good luck finding that measurement at the hardware store!

[C] Parallelogram | Parallelograms are the fair players of geometry. Their diagonals meet exactly in the middle, splitting each other perfectly in half like diplomatic mediators settling territorial disputes between opposite corners.

[C] 4/3 | Perpendicular slopes are negative reciprocals, like mathematical frenemies. Flip the fraction and change the sign: -3/4 becomes 4/3. They meet at perfect right angles, creating geometric drama wherever they intersect.

[D] 6 | Regular hexagons are symmetry superstars with 6 lines of reflection. Three pass through opposite vertices, three through midpoints of opposite edges. Bees knew what they were doing choosing this perfectly balanced shape!

[A] Dilation | Dilations are geometry's zoom feature. They shrink or enlarge figures while keeping all angles intact, like resizing a photo without distorting your face. Everything scales proportionally in this mathematical makeover.

[B] 36π | Cone volume equals one-third of a cylinder's: (1/3)πr²h = (1/3) × π × 9 × 12 = 36π. Think of it as an ice cream cone that holds exactly one-third of what a cylindrical container would.

[C] 1440° | With 10 sides, use (10-2) × 180° = 1440°. Decagons pack serious angular real estate, enough degrees to make four complete rotations with some leftover for a victory spin!

[A] Alternate interior angles are congruent | When a transversal crosses parallel lines, alternate interior angles match perfectly like synchronized swimmers. They're on opposite sides of the transversal but between the parallel lines, creating geometric harmony.

[C] 96 | Six square faces, each measuring 4×4 = 16, gives us 6×16 = 96. It's like gift-wrapping a perfectly cubic present where every face demands equal attention and wrapping paper.

[D] All sides are equal | A rhombus is democracy in action for quadrilaterals. Every side gets equal length, though angles might vary. Think of it as a square that's been gently pushed sideways but maintains its egalitarian side policy.

[A] 10 | Using the Pythagorean theorem: √(36+64) = √100 = 10. This creates a perfect 6-8-10 right triangle, a scaled-up version of the famous 3-4-5 triple that ancient builders loved.

[D] −2 | Parallel lines are slope twins separated at birth. They share identical steepness but live at different y-intercepts. The slope -2 means descending 2 units for every 1 unit rightward.

[A] 45 | Half of base times height: (1/2) × 10 × 9 = 45. Triangles are economical shapes, using exactly half the space a rectangle would need with the same dimensions.

[B] 5 | Two triangular ends plus three rectangular sides equals five faces total. It's like a mathematical sandwich with triangular bread and three rectangular fillings holding the geometric structure together.

[D] 7π/2 | A 90° angle captures one-quarter of the circle. With circumference 14π, the arc length is 14π/4 = 7π/2. It's like taking a perfectly measured slice from a circular pie.

[A] 36π | Using V = (4/3)πr³: (4/3) × π × 27 = 36π. Spheres pack maximum volume into minimum surface area, making them nature's most efficient containers (just ask bubbles and planets).

[A] (6, −3) | Multiply each coordinate by the scale factor: (2×3, -1×3) = (6, -3). It's like using a mathematical magnifying glass centered at the origin, tripling distances while preserving directions.

[C] Regular pentagon | Pentagons are the rebels of tessellation. Their 108° angles don't divide evenly into 360°, leaving gaps. Only triangles, squares, and hexagons can tile infinitely without overlaps or spaces.