Geometry & Measurement — Draw the picture. Then look for symmetry.

Two fundamental quantities of a 2D shape:

• Perimeter = the total length of the boundary. Walk around the shape; add up the steps.
• Area = how much surface is enclosed. Counted in square units.

One huge trap. The height in the triangle, parallelogram, and trapezoid formulas is always the perpendicular distance from the base to the opposite vertex (or opposite side). Using a slanted side instead of the perpendicular is one of the most common AMC mistakes.

When a region is too irregular for a single formula, cut it into pieces you DO know formulas for — rectangles, triangles, half-circles. Compute each piece, then ADD.

Often there’s more than ONE good cut. Both methods give the same answer:

A triangle inside a rectangle — cut differently

A pentagon with vertices (0,0), (4,0), (5,3), (2,5), (0,3). Cut into a rectangle and two triangles:

• Bottom rectangle: width 4, height 3 → area 12.
• Bottom-right triangle: base 1, height 3 → ½ · 1 · 3 = 1.5.
• Top triangle: base 5, height 2 → ½ · 5 · 2 = 5.

Total: 12 + 1.5 + 5 = 18.5 square units.

This is the single most common decomposition pattern on the AMC.

This works when the shaded region is defined by what's removed, not by what's added. Examples:

• A square with a circle removed: area = square − circle.
• A picture inside a frame: frame area = big rectangle − inner picture rectangle.
• The leftover after cutting: original area − cut piece.
• Two overlapping shapes: union area = A + B − intersection (inclusion-exclusion).

You never measure the shaded boundary directly. You measure the big rectangle and the hole, then subtract.

The reason this works: when a shape has an irregular boundary but is clearly 'big minus small', you don't need to compute the irregular boundary at all. You just compute the two enclosing shapes.

For any right triangle with legs a, b and hypotenuse c (the longest side, opposite the right angle):

This is the most famous equation in elementary geometry. It says: in a right triangle, the area of a square built on the hypotenuse equals the sum of the areas of squares built on the two legs.

Pythagorean triples are right triangles whose three sides are all whole numbers. Memorize these:

The three most common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.

Other triples worth knowing:

• 7-24-25
• 9-40-41
• 20-21-29 (rare but does show up)
• Any multiple of a triple is also a triple: 6-8-10, 9-12-15, 30-40-50, 10-24-26, etc.

If you see a right triangle with TWO of these numbers showing, you know the third without computing.

Special non-integer right triangles. These are also crucial:

• 45-45-90 (isosceles right triangle): legs equal, hypotenuse = leg · √2. Sides in ratio 1 : 1 : √2.
• 30-60-90: shorter leg = x, longer leg = x · √3, hypotenuse = 2x. Sides in ratio 1 : √3 : 2.

The 13-14-15 triangle — the one non-right triangle worth memorizing

It's not a right triangle, but its sides are all integers and so is its area:

Drop the altitude to the side of length 14. It splits into a 5-12-13 right triangle and a 9-12-15 right triangle (a 3-4-5 tripled). Both right triangles share the altitude 12, and the bases add to 5 + 9 = 14 ✓.

If you see a triangle on the AMC with sides 13, 14, 15 (or scalings of them), you can read off the area immediately.

If a figure has a line of symmetry or rotational symmetry, you can find one piece and use the symmetry to handle the others without re-computing.

Lines of symmetry (axes you can fold along):

• A square has 4 axes (two through midpoints, two diagonals).
• An equilateral triangle has 3 axes.
• A rectangle has 2 axes (through midpoints, not the diagonals).
• A regular hexagon has 6 axes.
• A circle has infinitely many — any line through the center.

Rotational symmetry: a figure looks the same after rotating by some angle. A square has order 4 (looks the same after 90°, 180°, 270°, 360°). A regular hexagon has order 6.

When a figure is shaded in a symmetric pattern, the shaded area is some clean fraction of the whole — often 1/2, 1/4, 3/8. Don't compute the irregular boundary if you can guess the fraction.

For 3D shapes:

A net is a 2D unfolding of a 3D shape — the pattern of faces you'd cut out of paper and fold up. A cube has 11 distinct nets (counting rotations/reflections as the same). The most common net looks like a plus-sign cross with 6 squares.

The key skill: given a net, identify which faces become opposite when folded.

Fold F up as the front of the cube. T folds up to the top, B folds down to the bottom (so T ↔ B). L folds to the left, R folds to the right (so L ↔ R). K wraps around to become the back (so F ↔ K).

Angle chasing is the geometry game where you keep applying angle rules to propagate known angles around a figure until you find the angle you need.

The exterior-angle theorem is one of the most-used rules. Picture a triangle ABC, and extend one side past vertex B to make an exterior angle. That exterior angle equals the sum of the two non-adjacent interior angles (A and C):

Start from what's labeled. Apply one rule. Label the new angle. Repeat. Most angle problems collapse in 3–4 steps.

Circles are common on AMC 8. The formulas:

The constant π ≈ 3.14159… For AMC, you can usually leave π in the answer; the multiple-choice options will too. Sometimes you'll need to approximate; π ≈ 22/7 or π ≈ 3.14 both work.

Common AMC moves with circles:

• Quarter circle (90° sector) area: π r²/4. Half circle area: π r²/2.
• Square inscribed in a circle of radius r: the diagonal of the square equals 2r, so the side equals r√2.
• Circle inscribed in a square of side s: the diameter equals s, so the radius equals s/2.
• For two circles tangent externally, distance between centers = R + r (sum of radii). For tangent internally (one inside the other), distance between centers = |R − r|.

Two circle facts that unlock a LOT of AMC problems

This means: every triangle drawn on a semicircle (with the diameter as one side) is a right triangle. The right angle sits on the curve.

Wherever a problem says “tangent to the circle at point P,” drop the radius to P and mark a right angle. The right angle usually unlocks a Pythagorean triangle nearby.

When a polygon's vertices are given as coordinates, the cleanest area formula is Shoelace.

How to apply it (for a triangle with vertices A=(0,0), B=(4,0), C=(0,3)):

1. Write coordinates in a column, repeating the first at the bottom.
2. Multiply down-right (x₁·y₂, x₂·y₃, x₃·y₁) and sum: 0·0 + 4·3 + 0·0 = 12.
3. Multiply down-left (y₁·x₂, y₂·x₃, y₃·x₁) and sum: 0·4 + 0·0 + 3·0 = 0.
4. Subtract, take absolute value, divide by 2: |12 − 0| / 2 = 6.

(Matches the formula ½ · base · height = ½ · 4 · 3 = 6 ✓.)

Shoelace works for ANY polygon (even concave ones). It's mostly used on triangles and quadrilaterals on AMC.

Pick’s Theorem — the magical dot-counting formula

Here’s a totally different way to find the area of a polygon when all its corners sit on grid dots (called lattice points). Instead of coordinates and arithmetic, you just count two kinds of dots.

Let’s try it. Below is a polygon drawn on a grid. Count dots, then plug in.

● boundary dots (B = 8)  ·  ● interior dots (I = 5)

Plug in: Area = 5 + 8/2 − 1 = 5 + 4 − 1 = 8. Done — no coordinates, no shoelace, just dot-counting.

When to use Shoelace vs Pick’s:

• If the polygon’s corners are easy coordinates (and especially if not on grid dots) → Shoelace.
• If the polygon sits on a clean grid where you can SEE and COUNT the dots → Pick’s.
• Both will always give the same answer. Use whichever is faster for that specific picture.

Folding problems: a paper or shape is folded along a line. After the fold, some parts overlap; some cuts go through multiple layers.

Key insights for folds:

• The fold line is an axis of symmetry. Each side mirrors the other across it.
• A point at distance d from the fold line maps to a point at distance d on the other side.
• A single cut through folded paper creates symmetric cuts when the paper is unfolded.

Rolling problems: a shape rolls around the outside of another shape. The total rotation has two parts:

• Rolling component = (perimeter traveled along) ÷ (perimeter of the rolling shape) × 360°.
• Corner-pivot component = at each corner of the fixed shape, the rolling shape pivots through the exterior angle of that corner. For a hexagon (interior 120°), each exterior angle is 60°, so 6 corners contribute 6·60° = 360°.

Add the two components for the total rotation.

Reflection: reflecting a point across a vertical line flips its x-coordinate. Across a horizontal line, flips y-coordinate. The shape's orientation reverses (left ↔ right mirror image).

Two shapes are similar if they have the same shape but possibly different sizes. Every corresponding angle is equal, and every pair of corresponding sides is in the same ratio. Think of them as enlarge-or-shrink copies of each other.

Walkthrough. Two squares are similar (all squares are, by definition). One has side 3, the other side 6. So the scale factor is k = 2.

• Perimeters: 12 and 24 — ratio 2.
• Areas: 9 and 36 — ratio 4 = 2².

This squaring of the area ratio is the most-used fact. A scale model that's half the size has quarter the surface area; a model that's one-tenth the size has one-hundredth the surface area.

Sides double → area quadruples. Always 2² = 4, never 2.

When are two triangles similar?

You don't need to check all six conditions (3 angles + 3 side ratios). Any one of these three is enough:

The AA trick is the workhorse. Whenever you spot two triangles sharing an angle (or formed by parallel lines crossed by a transversal — equal alternate-interior angles), they're similar by AA. From there, set up the side-ratio equation and solve.

Classic configuration: parallel cuts on a triangle. If a line parallel to one side of a triangle cuts the other two sides, it creates a smaller triangle similar to the original.