Geometry curriculum

Two angles are complementary when they add to 90°, as the two acute angles of a right triangle do. An angle of 28.5° has a complement of 90° − 28.5° = 61.5°. The usual slip is subtracting from 180° instead, which gives the supplement. Remember "complementary" as the corner of a square: 90°.

Two angles are supplementary when they add to 180°, like the two angles on either side of a line. An angle of 117.5° has a supplement of 180° − 117.5° = 62.5°. Any two angles that form a straight line together are supplementary, which is the fact most angle problems actually use.

When two lines cross they form four angles. Opposite angles, called vertical angles, are equal, and neighboring angles are supplementary. If one angle is 74.5°, the angle opposite it is also 74.5°, and each neighbor is 180° − 74.5° = 105.5°. Answering with the neighbor instead of the opposite angle is the common error.

The three angles of any triangle add to 180°. If two angles are 47.5° and 68°, the third is 180° − 47.5° − 68° = 64.5°. In a right triangle the two acute angles therefore add to 90°, and in an isosceles triangle the two base angles are equal. An angle sum other than 180° means a misread angle.

An inscribed angle, with its vertex on the circle, is half the central angle that cuts off the same arc. If the central angle is 150°, the inscribed angle on the same arc is 75°. A special case: an angle inscribed in a semicircle is always 90°, because the central angle is a straight 180°.

Perimeter is the distance around a shape, a length; area is the space it covers, in square units. A 17 by 9 rectangle has perimeter 2 × (17 + 9) = 52 and area 17 × 9 = 153. Choices often include the wrong one of the pair, so check whether the question asks for fencing (perimeter) or flooring (area).

A square's area is its side times itself: a side of 16 gives an area of 256 square units, while its perimeter is 4 × 16 = 64. Doubling the side quadruples the area, because both dimensions double. Using 4 times the side as the area confuses the perimeter formula with the area formula.

A triangle's area is half the base times the height, where the height is measured perpendicular to the base. A triangle with base 15 and height 8 has area ½ × 15 × 8 = 60. Forgetting the half gives 120, and using a slanted side as the height gives a wrong value; the height must meet the base at a right angle.

An equilateral triangle of side s has height (√3/2)s, so its area is ½ × s × (√3/2)s = (√3/4)s². With side 14, the area is (√3/4) × 196 = 49√3. Using s²/2, as if the height equaled the side, overstates the area, because the height is shorter than a side.

A regular hexagon is six equilateral triangles meeting at its center, each with the hexagon's side length. With side 12, each triangle has area (√3/4) × 144 = 36√3, so the hexagon has area 6 × 36√3 = 216√3, which matches the formula (3√3/2)s². Splitting a figure into familiar shapes is the general method.

In a right triangle, the squares of the legs add to the square of the hypotenuse: a² + b² = c². With legs 20 and 21, c² = 400 + 441 = 841, so c = 29. Adding the legs, to 41, is the classic error; the hypotenuse is always the longest side, but shorter than the sum of the legs.

Some whole-number right triangles recur: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples. Legs of 21 and 28 are 7 times 3 and 4, so the hypotenuse is 7 × 5 = 35. Spotting the pattern avoids squaring and taking roots, but check that the numbers really share the same multiple.

When the hypotenuse and one leg are known, subtract: b² = c² − a². With hypotenuse 25 and leg 15, b² = 625 − 225 = 400, so b = 20. Adding the squares instead, as if the unknown were the hypotenuse, gives a third side longer than the hypotenuse, which cannot be right.

A 45-45-90 triangle has two equal legs, and its hypotenuse is a leg times √2. A leg of 13 gives a hypotenuse of 13√2. Going the other way, a hypotenuse of 10 gives legs of 10/√2 = 5√2. Mixing this up with the 30-60-90 ratio, which uses √3, is the usual error.

In a 30-60-90 triangle the shortest side, opposite 30°, is x; the side opposite 60° is x√3; and the hypotenuse is 2x. With a shortest side of 11, the other sides are 11√3 and 22. Identify which side you have before scaling, since the hypotenuse is twice the shortest side, not twice the middle one.

The midpoint of a right triangle's hypotenuse is equally far from all three vertices, so the median from the right angle to the hypotenuse is half the hypotenuse. In a right triangle with legs 12 and 35 and hypotenuse 37, that median is 37 ÷ 2 = 18.5. It is the circle through all three vertices that makes this work.

For a right triangle with legs a and b and hypotenuse c, the radius of the inscribed circle is (a + b − c) ÷ 2. With legs 20 and 21 and hypotenuse 29, the radius is (20 + 21 − 29) ÷ 2 = 6. Half the hypotenuse is the radius of the circle through the vertices, not the inscribed one, and is a common trap answer.

The diameter is twice the radius, and circle formulas are written with the radius. A circle with a diameter of 18 has a radius of 9, so its area is 81π, not 324π. Before substituting into any formula, check which measurement the question gives; confusing them is the most common circle error.

The circumference, the distance around a circle, is 2πr, or πd. A circle of radius 13 has circumference 26π. Answers stay in terms of π unless the question asks for a decimal. Using πr², the area formula, for the distance around is the typical mix-up between the two circle formulas.

A circle's area is πr²: with radius 13 the area is 169π. The radius is squared, not doubled, so doubling the radius quadruples the area. Leave the answer in terms of π when the choices do, and convert to a decimal only when the question asks for one, using 3.14 for π.

An arc is a fraction of the circumference equal to its central angle over 360°. In a circle of radius 10, an arc with a 72° central angle is 72/360 = 1/5 of 20π, which is 4π. Using the area instead of the circumference, or forgetting the fraction, are the usual errors.

A sector is a fraction of the circle's area equal to its central angle over 360°. In a circle of radius 10, a 72° sector is 1/5 of 100π, which is 20π. Compare with the arc of the same angle, 4π: an arc is a length taken from the circumference, while a sector is an area taken from the whole disk.

A rectangular box has volume length × width × height: a 7 × 9 × 12 box holds 756 cubic units. Volume is in cubic units because it counts unit cubes. Adding the dimensions, or confusing volume with the area of one face, are the common errors.

Surface area adds the areas of all six faces, which come in three matching pairs: 2(lw + lh + wh). For a 7 × 9 × 12 box, that is 2(63 + 84 + 108) = 510 square units. Counting each pair only once, to 255, or using the volume, are the usual mistakes.

Any prism or cylinder has volume equal to its base area times its height. A cylinder with radius 5 and height 12 has volume π × 5² × 12 = 300π. If the question gives the diameter, halve it first: a diameter of 10 is a radius of 5, and using 10 as the radius would give 1,200π.

The longest straight line inside a box runs corner to corner, and the Pythagorean theorem extends to three dimensions: d = √(l² + w² + h²). For an 8 × 9 × 12 box, d = √(64 + 81 + 144) = √289 = 17. Using only two dimensions gives a face diagonal, which is shorter.

If every length is multiplied by k, areas multiply by k² and volumes by k³. Making every dimension of a solid 6 times larger makes its surface area 36 times larger and its volume 216 times larger. Multiplying the volume by k alone ignores that all three dimensions grew.

The distance between two points is the hypotenuse of a right triangle whose legs are the horizontal and vertical changes, dx and dy, so d = √(dx² + dy²). From (−4, 6) to (16, 27), dx = 20 and dy = 21, so d = √(400 + 441) = 29. Adding the changes, to 41, gives the path along the grid, not the straight-line distance.

The midpoint of a segment averages the coordinates: from (−7, 4) to (3, −10), it is ((−7 + 3) ÷ 2, (4 − 10) ÷ 2) = (−2, −3). Subtracting the coordinates instead of adding them finds half the change, not the middle point. Signs matter here, so keep negatives in parentheses.

Slope is the change in y divided by the change in x, taken in the same order for both. From (2, 5) to (8, −7), the rise is −7 − 5 = −12 and the run is 8 − 2 = 6, so the slope is −2. Mixing the order, subtracting the first y but the second x, flips the sign.

Move one vertex to the origin by subtracting its coordinates from the other two. If those two points are then (a, b) and (c, d), the area is half the absolute value of ad − bc. For vertices (0, 0), (8, 2), and (3, 7), the area is |8 × 7 − 3 × 2| ÷ 2 = 50 ÷ 2 = 25. Enclosing the triangle in a rectangle and subtracting the corner triangles gives the same answer.

Similar figures have the same shape, so every pair of corresponding sides has the same ratio. If two similar triangles have sides 4 and 10 in matching positions, the scale factor is 10/4 = 2.5, and a side of 12 in the small one corresponds to 30 in the large one. Match sides by position, not by length order.

Areas of similar figures scale with the square of the side ratio. If two similar triangles have sides in the ratio 2 : 5, their areas are in the ratio 4 : 25, so a small area of 12 becomes 12 × 25/4 = 75. Using the side ratio itself for areas, to get 30, is the classic error.

A polygon with n sides can be cut into n − 2 triangles from one vertex, so its interior angles add to (n − 2) × 180°. A 15-sided polygon's angles sum to 2,340°, and if it is regular each angle is 2,340° ÷ 15 = 156°. Dividing 360° by n gives an exterior angle, not an interior one.

The exterior angles of any convex polygon add to 360°, so a regular polygon's exterior angle is 360° ÷ n and its interior angle is 180° minus that. For 15 sides, the exterior angle is 24° and the interior angle is 156°, matching the interior-sum method with less arithmetic.

Each vertex of an n-sided polygon connects by a diagonal to every vertex except itself and its two neighbors, giving n − 3 diagonals per vertex. Each diagonal joins two vertices, so the total is n(n − 3) ÷ 2. A 15-sided polygon has 15 × 12 ÷ 2 = 90 diagonals. Forgetting to halve double-counts every diagonal.

Lengths are in units, areas in square units, and volumes in cubic units. An answer to a volume question in square units, or a perimeter in square units, signals a formula mix-up. Converting between units also scales by the square or cube: 1 square foot is 144 square inches, not 12.

A quick sketch with every given length and angle labeled shows which formula applies, which side is the hypotenuse, and which height is perpendicular to which base. Many errors come from using a slanted side as a height or the diameter as a radius, and a labeled sketch makes those visible.

Use simple facts as checks. The hypotenuse is the longest side of a right triangle but shorter than the sum of the legs. A triangle's angles add to 180°. A sector cannot have more area than its circle, and a box's face diagonal is shorter than its space diagonal. An answer that breaks one of these is wrong.