Geometry questions test a short list of facts used carefully: angle sums, area and perimeter, right triangles, circles, solids, coordinates, and similar figures. Most wrong answers come from using the right formula on the wrong measurement: a diameter where the formula wants a radius, a slanted side where it wants a height, a side ratio where it wants an area ratio. The chapters cover each kind of fact and end with the checks that catch those slips.
Each chapter opens with the short version. Tap one to read the detail.
Angles, triangles, and polygons
~2 min
Angles on a straight line add to 180°, two angles that make a right angle add to 90°, and a triangle's angles add to 180°. Vertical angles are equal, an inscribed angle is half the central angle on the same arc, and a polygon's angles follow from the triangles inside it.
Most angle questions rest on three sums. Angles that together make a straight line add to 180°, so an angle of 133.5° leaves 46.5° beside it. Two angles that make a right angle add to 90°. The three angles of a triangle add to 180°: with two angles of 52.5° and 71°, the third is 56.5°. Where two lines cross, the opposite angles are equal and the neighboring ones add to 180°.
In a circle, an angle with its vertex on the circle is half the central angle that stands on the same arc, so a central angle of 164° gives an inscribed angle of 82°. An angle inscribed in a semicircle is therefore always 90°.
For a polygon with n sides, the interior angles add to (n − 2) × 180°, because n − 2 triangles fill it from one vertex. The exterior angles of any convex polygon add to 360°, so a regular polygon's exterior angle is 360° ÷ n. For 18 sides that is 20°, and each interior angle is 160°. Each vertex sends a diagonal to every vertex except itself and its two neighbors, so there are n(n − 3) ÷ 2 diagonals: 135 for 18 sides.
Rule: find the sum the angles must make (90°, 180°, 360°, or (n − 2) × 180°), then subtract what you know.
Perimeter and area
~2 min
Perimeter is a length around a shape; area is the surface it covers, in square units. A triangle's area is half of base times perpendicular height, and unfamiliar shapes split into familiar ones.
Perimeter adds the lengths around a shape, and area counts the square units inside it. An 11 by 23 rectangle has perimeter 2 × (11 + 23) = 68 and area 11 × 23 = 253. Both numbers tend to appear among the choices, so check which one the question asks for.
A triangle's area is half the base times the height, and the height must be perpendicular to the base. With base 26 and height 9 the area is 117. Forgetting the half doubles the answer, and using a slanted side as the height inflates it.
Shapes you have no formula for usually split into ones you do. An equilateral triangle of side s has height (√3/2)s, so its area is (√3/4)s²: for a side of 16, that is 64√3. A regular hexagon is six equilateral triangles meeting at its center, so a hexagon of side 16 has area 6 × 64√3 = 384√3. An L-shaped room is two rectangles, and a rectangle with a corner notch cut out is the full rectangle minus the notch.
Areas scale with the square of length. Doubling a square's side quadruples its area, and a square foot is 12 × 12 = 144 square inches, not 12.
Rule: name the measurement (the length around or the surface inside), use a perpendicular height, and split unfamiliar shapes into familiar ones.
Right triangles
~2 min
The squares of the legs add to the square of the hypotenuse. Recognize scaled triples, subtract squares for a missing leg, and use the fixed side ratios of the 45-45-90 and 30-60-90 triangles.
In a right triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle. With legs 33 and 56, c² = 1,089 + 3,136 = 4,225, so c = 65. With a hypotenuse of 61 and one leg of 11, the other leg is √(3,721 − 121) = √3,600 = 60. The hypotenuse is always the longest side, yet shorter than the two legs added together, which rules out many trap choices at a glance.
Some right triangles have whole-number sides and recur in scaled forms: 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Legs of 27 and 36 are 9 times 3 and 4, so the hypotenuse is 9 × 5 = 45.
Two special triangles have fixed ratios. A 45-45-90 triangle has equal legs, and its hypotenuse is a leg times √2, so a leg of 17 gives 17√2. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so a shortest side of 14 gives the other sides 14√3 and 28.
Two facts about a right triangle's circles also come up. The midpoint of the hypotenuse is the same distance from all three corners, so the median from the right angle is half the hypotenuse. The radius of the inscribed circle is (a + b − c) ÷ 2: for legs 33 and 56 and hypotenuse 65, that is 12.
Rule: identify the hypotenuse first, look for a scaled triple, and use the fixed ratios of the 45-45-90 and 30-60-90 triangles.
Circles, arcs, and sectors
~2 min
Check whether you have the radius or the diameter. Circumference is 2πr and area is πr², and an arc or a sector is the share of the circle set by its central angle.
Circle formulas use the radius, and questions often give the diameter. A circle 34 across has a radius of 17, so its area is 289π; squaring the diameter instead gives 1,156π, four times too much. Its circumference is 2π × 17 = 34π.
An arc is a piece of the circumference and a sector is a piece of the area, each the share of the circle given by its central angle over 360°. In a circle of radius 9, a 40° central angle is 40/360 = 1/9 of the circle, so its arc is 1/9 of 18π, which is 2π, and its sector is 1/9 of 81π, which is 9π. Mixing up which total the share applies to is the common slip: an arc is a length, and a sector is an area.
Leave answers in terms of π when the choices are written that way, and use 3.14 for π only when a decimal is asked for.
Rule: convert to the radius first, then take the central angle's share of the circumference for an arc or of the area for a sector.
Solids and similar figures
~2 min
Volume multiplies three dimensions, surface area adds the faces, and a box's longest diagonal extends the Pythagorean theorem. Scaling every length by k multiplies areas by k² and volumes by k³, and similar figures compare the same way.
A box's volume is length × width × height, in cubic units: a 6 × 11 × 13 box holds 858 cubic units. Its surface area adds all six faces, which come in three matching pairs: 2 × (66 + 78 + 143) = 574 square units. A cylinder's volume is its circular base times its height, so radius 7 and height 10 give 490π. Halve a given diameter before using it.
The longest straight line inside a box runs corner to corner, and its length is √(l² + w² + h²). For a 2 × 10 × 11 box, that is √(4 + 100 + 121) = √225 = 15.
When every length of a figure is multiplied by k, its areas multiply by k² and its volumes by k³. A model built at one third of the real size has 1/9 of the real surface area and 1/27 of the volume. Similar figures work the same way: corresponding sides share one ratio, and areas use its square. Two similar triangles with sides in the ratio 3 : 7 have areas in the ratio 9 : 49, so a small area of 18 becomes 98. Using the side ratio for the areas, to get 42, is the classic error.
Rule: multiply three lengths for volume, add the faces for surface area, and square or cube the scale factor for areas and volumes.
Coordinates and final checks
~2 min
Distance is the Pythagorean theorem on the grid, a midpoint averages the coordinates, and slope is rise over run taken in one consistent order. Before choosing, check the units, sketch the figure, and test the answer against basic geometric facts.
On a coordinate grid, the distance between two points is the hypotenuse of a right triangle whose legs are the horizontal and vertical changes. From (−3, 2) to (21, 9), the changes are 24 and 7, so the distance is √(576 + 49) = √625 = 25. Adding the changes, to 31, measures a path along the grid lines instead.
The midpoint averages the coordinates: from (−9, 5) to (5, −13), it is (−2, −4). Slope is the change in y over the change in x, taken in the same order for both: from (1, 8) to (6, −12), the slope is −20 ÷ 5 = −4. To find a triangle's area from its vertices, move one vertex to the origin; if the other two are then (a, b) and (c, d), the area is half of |ad − bc|.
Then check. Lengths are in units, areas in square units, and volumes in cubic units, so an answer in the wrong unit signals a formula mix-up. A quick labeled sketch shows which side is the hypotenuse and which height meets which base. Simple facts rule out trap answers: a triangle's angles add to 180°, the hypotenuse is the longest side, and a sector cannot hold more area than its circle.
Rule: treat coordinate distance as a right triangle, keep subtraction in one consistent order, and test every answer against its units and basic geometric facts.
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