LET Elementary Mathematics — Plane and Solid GeometryRevision Notes

Formulas

Exam Tips

• Memorize the C-S-V pattern: Complementary = 90°, Supplementary = 180°, Vertical = equal.
• When a transversal problem gives you one angle, use: corresponding → equal; alternate interior → equal; co-interior → subtract from 180°.
• Always label angles before computing — draw the figure if allowed.

Key Points

• An angle is measured in degrees (°). The full rotation is 360°.
• Acute angle: less than 90°. Right angle: exactly 90°. Obtuse angle: between 90° and 180°. Straight angle: exactly 180°. Reflex angle: between 180° and 360°.
• Complementary angles: two angles whose sum is 90°. Example: 35° and 55° are complementary.
• Supplementary angles: two angles whose sum is 180°. Example: 110° and 70° are supplementary.
• Vertical angles (vertically opposite angles): formed when two lines intersect. They are always EQUAL.
• Adjacent angles on a straight line form a linear pair and are supplementary.
• When a transversal crosses two PARALLEL lines: (1) Corresponding angles are EQUAL. (2) Alternate interior angles are EQUAL. (3) Co-interior (same-side interior) angles are SUPPLEMENTARY (sum to 180°).
• LET TIP: Parallel-line transversal problems almost always give one angle and ask for another. Identify the relationship first (corresponding, alternate, co-interior), then apply the rule.

Definitions

Section Title

Angles and Angle Pairs

Common Mistakes

• Confusing complementary (90°) with supplementary (180°) — remember: 'C' comes before 'S' in the alphabet, and 90° comes before 180°.
• Assuming alternate exterior angles are supplementary — they are actually EQUAL (like alternate interior angles).
• Forgetting that vertical angles are equal only when formed by TWO straight lines crossing — not applicable to multiple intersecting lines.

Formulas

Exam Tips

• For 'can these form a triangle?' items: add the two SMALLER sides and check if the sum is GREATER than the largest side.
• For isosceles triangle problems: identify the vertex angle first; base angles = (180° − vertex angle) ÷ 2.
• For similarity problems involving shadows or heights: set up a proportion and cross-multiply.
• The AA criterion is the easiest similarity test: if two angles are equal, the triangles are similar — use it first.

Key Points

• The interior angles of ANY triangle always sum to 180°.
• Classification by sides: Equilateral (all 3 sides equal, each angle = 60°), Isosceles (2 sides equal, base angles equal), Scalene (no sides equal).
• Classification by angles: Acute (all angles < 90°), Right (one angle = 90°), Obtuse (one angle > 90°).
• Exterior angle theorem: An exterior angle of a triangle equals the SUM of the two NON-ADJACENT (remote) interior angles.
• Triangle inequality: The sum of any two sides must be GREATER than the third side. Example: sides 3, 4, 8 CANNOT form a triangle because 3 + 4 = 7 < 8.
• Congruence postulates (same shape AND size): SSS, SAS, ASA, AAS, HL (for right triangles only).
• Similarity criteria (same shape, proportional sizes): AA, SSS-proportional, SAS-proportional.
• In SIMILAR triangles: corresponding angles are EQUAL and corresponding sides are in the SAME RATIO.
• Areas of similar figures scale by the SQUARE of the ratio of corresponding sides. If sides are in ratio 1:3, areas are in ratio 1:9.
• Base angles of an isosceles triangle are EQUAL.

Definitions

Section Title

Triangles – Classification, Congruence, and Similarity

Common Mistakes

• Using HL congruence for non-right triangles — HL is ONLY for right triangles.
• Confusing congruence (SSS, SAS, ASA, AAS, HL) with similarity (AA, SSS-proportional, SAS-proportional).
• Forgetting that the area ratio is k-SQUARED (not k) for similar figures.
• In isosceles triangles, mixing up the vertex angle (between the equal sides) and the base angles (equal to each other).

Formulas

Exam Tips

• Before computing, check if the numbers are a Pythagorean triple or a multiple of one — saves time.
• For 'is this a right triangle?' items: square all three sides, add the two SMALLER squares, and check if they equal the LARGEST square.
• Distance formula items on coordinate geometry are just the Pythagorean theorem in disguise — treat the horizontal and vertical differences as the two legs.

Key Points

• The Pythagorean theorem applies ONLY to RIGHT triangles: a² + b² = c², where c is the HYPOTENUSE (the longest side, opposite the right angle).
• Memorize these Pythagorean triples and their multiples: 3-4-5 (×2: 6-8-10; ×3: 9-12-15), 5-12-13, 8-15-17.
• Recognizing triples saves computation time on the LET — no square roots needed.
• Converse: If a² + b² = c², then the triangle IS a right triangle.
• To find the HYPOTENUSE: c = √(a² + b²).
• To find a LEG: a = √(c² − b²).
• Common application: distance problems (ladder against wall, diagonal of a rectangle, distance between two points on a grid).

Definitions

Section Title

The Pythagorean Theorem

Common Mistakes

• Putting one of the LEGS as c — always identify the hypotenuse (opposite the right angle) first.
• Not recognizing multiples of Pythagorean triples (e.g., not seeing that 9-12-15 is just 3-4-5 × 3).
• Using a² + b² = c² to find a leg: you must rearrange to a = √(c² − b²), not √(c² + b²).
• Applying the Pythagorean theorem to non-right triangles.

Formulas

Exam Tips

• Quick check table — memorize: Triangle 180°, Quadrilateral 360°, Pentagon 540°, Hexagon 720°, Octagon 1080°.
• Exterior angles ALWAYS total 360° — use this as a shortcut for exterior-angle problems.
• 'Is every square a rhombus?' → YES. 'Is every rhombus a square?' → NO. Know the hierarchy.

Key Points

• A polygon is a closed plane figure with straight sides.
• Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
• Each interior angle of a REGULAR polygon (all sides and angles equal) = (n − 2) × 180° ÷ n.
• Sum of EXTERIOR angles of ANY convex polygon = 360° (always, regardless of n).
• Each exterior angle of a regular polygon = 360° ÷ n.
• Know the angle sums for common polygons: Triangle (n=3): 180°; Quadrilateral (n=4): 360°; Pentagon (n=5): 540°; Hexagon (n=6): 720°; Octagon (n=8): 1080°.
• Quadrilateral hierarchy: Square ⊂ Rectangle ⊂ Parallelogram; Square ⊂ Rhombus ⊂ Parallelogram; Trapezoid has exactly one pair of parallel sides.
• Every square is a rectangle and a rhombus, but not every rectangle is a square.

Definitions

Section Title

Polygons – Interior and Exterior Angles

Common Mistakes

• Using the per-angle formula for IRREGULAR polygons — it only works for regular (all-equal) polygons.
• Forgetting that the exterior angle sum is ALWAYS 360°, not (n−2)×180°.
• Thinking a rhombus must have right angles — a rhombus only requires four equal sides; right angles make it a SQUARE.
• Confusing 'each exterior angle' (360°÷n) with 'each interior angle' (supplementary to the exterior angle).

Formulas

Exam Tips

• Read the problem twice: 'how much fencing?' = perimeter; 'how much tile/paint?' = area.
• When r is a multiple of 7, use π = 22/7 — it cancels and avoids decimals.
• For composite figures, sketch and label each part separately before computing.
• For the trapezoid, the formula is the AVERAGE of the two parallel sides × height.

Key Points

• Perimeter is the total LENGTH around a figure — always in LINEAR units (m, cm, km).
• Area is the amount of SURFACE covered — always in SQUARE units (m², cm², km²).
• For composite figures: split into standard shapes, compute each area, then ADD (or subtract for cutouts).
• ALWAYS convert all lengths to the SAME unit before computing; mixing meters and centimeters causes errors by factors of 100 or 10,000.
• Use π ≈ 3.1416 for general problems, or π = 22/7 when the radius is a multiple of 7 (e.g., r = 7, 14, 21) — 22/7 cancels cleanly.
• Context clues for what to compute: 'fencing' or 'border' → PERIMETER; 'flooring,' 'painting,' or 'covering' → AREA; 'filling' or 'capacity' → VOLUME.

Definitions

Section Title

Perimeter and Area of Plane Figures

Common Mistakes

• Using the slant side as the HEIGHT in triangle/parallelogram/trapezoid area formulas — height must be PERPENDICULAR to the base.
• Forgetting to SQUARE the radius in the circle area formula (writing πr instead of πr²).
• Mixing units (e.g., length in meters and width in centimeters) before multiplying for area.
• Using the diameter instead of the radius in the area formula.
• Forgetting to divide by 2 in the trapezoid area formula.

Formulas

Exam Tips

• Cone and pyramid = 1/3 × (matching prism/cylinder formula). This 1/3 is tested repeatedly.
• Surface area problems: list all faces, compute each, then add — do not skip any face.
• Sphere: V and SA have a curious coincidence when r = 3: both ≈ 113.04 (with π ≈ 3.14). Know this for checking.
• Cubic units for volume: if the answer has m² or cm², it is WRONG for a volume question.

Key Points

• Surface area (SA) is the TOTAL area of all outer faces of a 3D solid — in SQUARE units.
• Volume (V) is the amount of space enclosed by a solid — in CUBIC units.
• A cone's volume is ONE-THIRD of the cylinder with the same base radius and height.
• A pyramid's volume is ONE-THIRD of the prism with the same base and height.
• The slant height (l) of a cone is NOT the same as the vertical height (h); they are related by l² = r² + h².
• Sphere formulas involve r³ (volume) and r² (surface area) — both use the same radius.
• Recognize context: 'how much water can the tank hold?' = VOLUME; 'how much paint to cover the tank?' = SURFACE AREA.

Definitions

Section Title

Surface Area and Volume of Solids

Common Mistakes

• Using vertical height (h) instead of slant height (l) in the LATERAL SURFACE AREA of a cone.
• Forgetting the factor of 1/3 for cone and pyramid volumes.
• Confusing surface area (square units) with volume (cubic units) in the final answer.
• Using diameter instead of radius in sphere or cylinder formulas.
• Forgetting to count BOTH circular bases of a cylinder in the surface area formula (2πr² + 2πrh).

Formulas

Exam Tips

• Given diameter d? Halve it to get r before plugging into any formula.
• Given circumference C? Solve 2πr = C for r first, then find area if needed.
• Use 22/7 when r = 7, 14, 21, 28 (multiples of 7) — it cancels and gives a whole number answer.
• For a composite figure with a semicircle on top of a rectangle: Area = (l × w) + (πr²/2); Perimeter = 2l + w + πr (replace top side of rectangle with arc).

Key Points

• Key parts: Center (O), Radius (r) — center to edge, Diameter (d = 2r) — across through center, Chord — joins two points on circle, Arc — part of circumference, Tangent — touches circle at exactly ONE point.
• A tangent is PERPENDICULAR to the radius drawn to the point of tangency.
• Central angle: vertex is at the CENTER; its measure equals the arc it intercepts.
• The circumference (perimeter of a circle) = πd = 2πr.
• If you know ANY ONE of radius, diameter, circumference, or area — you can find ALL the others.
• π (pi) ≈ 3.1416 is the universal ratio of any circle's circumference to its diameter.
• Semicircle area = πr² ÷ 2; semicircle perimeter = πr + 2r (arc + diameter).

Definitions

Section Title

Circles – Parts, Formulas, and Relationships

Common Mistakes

• Using d (diameter) in the area formula instead of r: writing πd² instead of πr².
• Forgetting to find r first when given the diameter or circumference.
• Confusing arc (part of the boundary) with chord (a straight line inside the circle).
• Computing a semicircle perimeter as πr only — forgetting to add the diameter (2r) for the straight edge.