Plane and Solid Geometry - Revision Notes
Geometry is one of the most consistently tested areas in the LET Mathematics component. For BEEd graduates preparing to teach Grades 1–6, mastery of plane and solid geometry is doubly important: you must solve LET items correctly AND understand how these concepts are taught under the K–12 Mathematics curriculum (DepEd). This chapter covers angles, triangles, polygons, circles, the Pythagorean theorem, perimeter, area, surface area, and volume. The winning approach is to memorize the formulas exactly, practice applying them step by step, and always check whether the question asks for length (linear units), area (square units), or volume (cubic units). Every formula in this chapter appears in actual LET items, so treat this review as both an exam tool and a teaching reference.
Angles and Angle Pairs
Formulas
Complementary: A + B = 90°
Example
One angle is 32.5°, its complement is 90° − 32.5° = 57.5°
Variables
A and B are two angle measures
Application
Find a missing angle when two angles together form a right angle
Supplementary: A + B = 180°
Example
One angle is 115°, its supplement is 180° − 115° = 65°
Variables
A and B are two angle measures
Application
Find a missing angle when two angles together form a straight line
Vertical angles: ∠1 = ∠3 and ∠2 = ∠4
Example
If ∠1 = 72°, then the vertically opposite ∠3 = 72°; adjacent ∠2 = 180° − 72° = 108°
Variables
Angles formed at the intersection of two lines
Application
Identify equal angles at an intersection without measuring
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
Transversal
A line that crosses two or more other lines at distinct points
Importance
Creates the corresponding, alternate interior, and co-interior angle pairs used in parallel-line problems
Alternate Interior Angles
Angle pairs on opposite sides of the transversal, between the parallel lines
Importance
These are EQUAL when the lines are parallel — a key fact in many LET items
Co-interior (Same-Side Interior) Angles
Angle pairs on the SAME side of the transversal, between the parallel lines
Importance
These are SUPPLEMENTARY (sum to 180°) when the lines are parallel
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.
Triangles – Classification, Congruence, and Similarity
Formulas
Angle sum: A + B + C = 180°
Example
If A = 40° and B = 70°, then C = 180° − 40° − 70° = 70° (isosceles triangle)
Variables
A, B, C are the three interior angles of the triangle
Application
Find a missing interior angle given the other two
Exterior angle: Ext = A + B (remote interior angles)
Example
Remote interior angles are 45° and 65°. Exterior angle = 45° + 65° = 110°
Variables
Ext is the exterior angle; A and B are the two non-adjacent interior angles
Application
Find an exterior angle or a missing interior angle
Similarity ratio: a/a' = b/b' = c/c' (= k, the scale factor)
Example
A 1.5 m student has a 2 m shadow; flagpole has 16 m shadow. 1.5/2 = h/16 → h = 12 m
Variables
a, b, c are sides of original; a', b', c' are sides of similar figure; k is scale factor
Application
Find an unknown side using proportional reasoning (cross-multiplication)
Area ratio: (Area₁/Area₂) = k²
Example
Two similar triangles have sides in ratio 1:3. If smaller area = 10 cm², larger = 10 × 9 = 90 cm²
Variables
k is the ratio of corresponding sides
Application
Find area of similar figure when the scale factor is known
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
Congruent Triangles
Triangles that are identical in both shape and size; all corresponding sides and angles are equal
Importance
Congruence proofs (SSS, SAS, ASA, AAS, HL) appear in LET reasoning items
Similar Triangles
Triangles with the same shape but different sizes; corresponding angles are equal and corresponding sides are proportional
Importance
Used in indirect measurement problems (shadow problems, map-scale problems) which are common in LET
Hypotenuse-Leg (HL) Theorem
Two RIGHT triangles are congruent if their hypotenuses and one pair of legs are equal
Importance
Only applies to RIGHT triangles — do not apply to other triangle types
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).
The Pythagorean Theorem
Formulas
a² + b² = c²
Example
Ladder foot is 6 m from wall, reaches 8 m up: c² = 6² + 8² = 36 + 64 = 100; c = 10 m (3-4-5 triple × 2)
Variables
a and b are the two legs (shorter sides); c is the hypotenuse (longest side, opposite the right angle)
Application
Find the missing side of a right triangle given the other two sides
Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Example
Distance from (1, 2) to (4, 6): d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5 units
Variables
x₁, y₁ are coordinates of point 1; x₂, y₂ are coordinates of point 2
Application
Find the straight-line distance between two points on the Cartesian plane
Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Example
Midpoint of (2, 3) and (8, 7): M = ((2+8)/2, (3+7)/2) = (5, 5)
Variables
x₁, y₁ and x₂, y₂ are coordinates of the two endpoints
Application
Find the center point of a line segment on the coordinate plane
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
Hypotenuse
The side of a right triangle opposite the right angle; always the LONGEST side
Importance
Always place c (not a or b) as the hypotenuse in the formula; mixing this up is the most common error
Pythagorean Triple
A set of three positive integers (a, b, c) that satisfy a² + b² = c², such as 3-4-5 or 5-12-13
Importance
Recognizing these triples instantly gives you the answer without computing square roots
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.
Polygons – Interior and Exterior Angles
Formulas
Sum of interior angles = (n − 2) × 180°
Example
Hexagon: (6 − 2) × 180° = 4 × 180° = 720°
Variables
n = number of sides of the polygon
Application
Find the total of all interior angles for any polygon
Each interior angle of regular polygon = (n − 2) × 180° ÷ n
Example
Regular hexagon: 720° ÷ 6 = 120° per angle
Variables
n = number of sides
Application
Find one interior angle of a regular (equilateral and equiangular) polygon
Sum of exterior angles = 360° (always)
Example
Regular octagon: each exterior angle = 360° ÷ 8 = 45°
Variables
Applies to any convex polygon
Application
Find a missing exterior angle or confirm a polygon is convex
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
Regular Polygon
A polygon with ALL sides equal (equilateral) AND all angles equal (equiangular)
Importance
The 'per-angle' formula only applies to REGULAR polygons; irregular polygons require adding all given angles
Convex Polygon
A polygon where all interior angles are less than 180° and no vertex points inward
Importance
The 360° exterior angle rule applies to CONVEX polygons
Parallelogram
A quadrilateral with two pairs of parallel sides; opposite sides are equal and opposite angles are equal
Importance
Parent figure for rectangles, rhombuses, and squares — knowing this hierarchy answers 'which is always true?' items
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).
Perimeter and Area of Plane Figures
Formulas
Square: Perimeter = 4s; Area = s²
Example
Square with s = 9 m: Perimeter = 36 m; Area = 81 m²
Variables
s = length of one side
Application
Fencing a square lot or finding the area of a square tile
Rectangle: Perimeter = 2(l + w); Area = l × w
Example
Rectangle 8 m × 5 m: Perimeter = 2(8 + 5) = 26 m; Area = 40 m²
Variables
l = length; w = width
Application
Most common formula on LET — used for rooms, lots, gardens
Triangle: Area = ½ × b × h
Example
Base = 12 m, height = 5 m: Area = ½ × 12 × 5 = 30 m²
Variables
b = base; h = perpendicular height (NOT slant side)
Application
Area of triangular lots or rooftop cross-sections
Parallelogram: Area = b × h
Example
Base = 10 cm, height = 6 cm: Area = 60 cm²
Variables
b = base; h = perpendicular height
Application
Area of tilted rectangular shapes
Trapezoid: Area = ½ × (a + b) × h
Example
Parallel sides 10 m and 6 m, height 4 m: Area = ½ × 16 × 4 = 32 m²
Variables
a and b = the two parallel sides; h = perpendicular height between them
Application
Area of trapezoidal lots or cross-sections of irrigation canals
Circle: Circumference C = 2πr = πd; Area = πr²
Example
r = 7 cm: C = 2 × (22/7) × 7 = 44 cm; Area = (22/7) × 49 = 154 cm²
Variables
r = radius; d = diameter = 2r
Application
Fencing a circular garden (circumference) or painting a circular floor (area)
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
Perimeter
The total distance around the boundary of a plane figure
Importance
Used for 'how much fencing/border/trim is needed?' — LINEAR units only
Area
The measure of the two-dimensional surface enclosed by a figure
Importance
Used for 'how much floor/paint/material is needed?' — always SQUARE units
Height (of a geometric figure)
The PERPENDICULAR distance from the base to the opposite vertex or side — NOT the slant side
Importance
The most common source of error in triangle and trapezoid area problems
Composite Figure
A shape made up of two or more standard geometric figures combined together
Importance
LET often shows L-shaped lots or rectangles with semicircular ends — decompose and add
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.
Surface Area and Volume of Solids
Formulas
Cube: V = s³; SA = 6s²
Example
Cube s = 4 cm: V = 64 cm³; SA = 6 × 16 = 96 cm²
Variables
s = edge length
Application
Volume of a cubic storage box or ice block
Rectangular Prism: V = l × w × h; SA = 2(lw + lh + wh)
Example
Box 8 × 5 × 3 cm: V = 120 cm³; SA = 2(40 + 24 + 15) = 158 cm²
Variables
l = length; w = width; h = height
Application
Capacity of a classroom aquarium or storage box
Cylinder: V = πr²h; SA = 2πr² + 2πrh
Example
r = 3 m, h = 10 m: V = 3.14 × 9 × 10 = 282.6 m³
Variables
r = radius of circular base; h = height
Application
Volume of a water drum or paint can
Cone: V = (1/3)πr²h; SA = πr² + πrl
Example
r = 3 cm, h = 4 cm: l = √(9+16) = 5 cm; V = (1/3)(3.14)(9)(4) ≈ 37.68 cm³
Variables
r = radius; h = vertical height; l = slant height = √(r² + h²)
Application
Volume of an ice cream cone or conical tank
Sphere: V = (4/3)πr³; SA = 4πr²
Example
r = 3 cm: V = (4/3)(3.14)(27) ≈ 113.04 cm³; SA = 4(3.14)(9) ≈ 113.04 cm²
Variables
r = radius
Application
Volume of a ball or spherical water tank
Pyramid: V = (1/3) × B × h
Example
Square base 6 × 6, height 4: V = (1/3)(36)(4) = 48 cubic units
Variables
B = area of the base; h = vertical height
Application
Volume of pyramid-shaped structures
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
Surface Area
The sum of the areas of all outer faces (flat and curved) of a three-dimensional solid
Importance
Used for 'painting,' 'wrapping,' or 'covering' problems — answer in square units
Volume
The measure of three-dimensional space enclosed within a solid
Importance
Used for 'filling,' 'capacity,' 'how much water' problems — answer in cubic units
Slant Height (of a cone)
The distance from the apex of the cone to any point on the circular edge of the base, measured along the surface
Importance
Used in the SURFACE AREA formula of a cone; different from vertical height; compute with Pythagorean theorem if not given
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).
Circles – Parts, Formulas, and Relationships
Formulas
Circumference: C = 2πr = πd
Example
r = 14 m: C = 2 × (22/7) × 14 = 88 m
Variables
r = radius; d = diameter
Application
Distance around a circular track or garden boundary
Area of circle: A = πr²
Example
r = 7 cm: A = (22/7) × 49 = 154 cm²
Variables
r = radius
Application
Area of a circular floor, plot, or table
Diameter: d = 2r
Example
If C = 44 cm, then 2πr = 44; r = 44 ÷ (2 × 22/7) = 44 × 7/44 = 7 cm
Variables
r = radius
Application
Convert between radius and diameter in any circle formula
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
Radius
The distance from the center of a circle to any point on its circumference
Importance
All circle formulas are written in terms of r — always find r first before computing
Tangent
A line that touches a circle at exactly one point and is perpendicular to the radius at that point
Importance
The perpendicularity of a tangent and radius is a common deduction in geometry proofs and LET items
Chord
A line segment joining any two points on a circle; the diameter is the longest possible chord
Importance
Distinguishing chord from radius and diameter prevents definition errors
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.
Connections
- The Pythagorean theorem is the foundation of the distance formula in coordinate geometry — both compute the hypotenuse of a right triangle formed by horizontal and vertical distances.
- Angle-sum rules for triangles (180°) connect to the polygon interior angle formula: a polygon of n sides can be divided into (n−2) triangles, each contributing 180°.
- Similar triangle ratios directly connect to real-world indirect measurement (shadow problems, scale maps, architectural drawings) — a key topic in Grade 5–6 K–12 Math.
- The area of a trapezoid formula is an extension of the parallelogram area: treating the trapezoid as an 'average-width' parallelogram gives ½(a+b)h.
- Cone and pyramid volumes are exactly one-third of their corresponding cylinder and prism — connecting flat-base area formulas to 3D volume.
- The circumference and area of a circle are linked through r: if you know C, you can find r, and then compute A = πr² — all three measures describe the SAME circle.
- Complementary and supplementary angle relationships reappear in parallel-line transversal problems (co-interior angles) and in right-triangle geometry (two acute angles of a right triangle are complementary).
- Surface area of a cylinder (2πr² + 2πrh) connects circle area (the two bases) and rectangle area (the lateral face 'unwrapped' = 2πr × h).
- Quadrilateral properties (parallelogram, rectangle, rhombus, square) form a hierarchy that connects algebraic conditions with geometric figures — essential for 'which statement is always true?' items.
- The 3-4-5 Pythagorean triple connects to the distance between coordinate points, the diagonal of a 3×4 rectangle, and the standard Grade 4–6 right-triangle problems in DepEd learning materials.
Quick review questions
Two angles are supplementary. One angle is three times the other. Find both angles.
Answer
45° and 135°
Let the smaller angle = x. Then the larger = 3x. Supplementary means they sum to 180°: x + 3x = 180° → 4x = 180° → x = 45°. The angles are 45° and 3 × 45° = 135°. Check: 45° + 135° = 180°. ✓
A right triangle has legs of 5 cm and 12 cm. What is the length of the hypotenuse?
Answer
13 cm
Recognize the 5-12-13 Pythagorean triple. Verify: 5² + 12² = 25 + 144 = 169 = 13². c = √169 = 13 cm.
Find the sum of the interior angles of a decagon (10-sided polygon).
Answer
1,440°
Sum = (n − 2) × 180° = (10 − 2) × 180° = 8 × 180° = 1,440°.
A circular garden has a radius of 7 m. Using π = 22/7, find its area and circumference.
Answer
Area = 154 m²; Circumference = 44 m
Area = πr² = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 m². Circumference = 2πr = 2 × (22/7) × 7 = 2 × 22 = 44 m.
A cylindrical water drum has a radius of 3 m and height of 10 m. Find its volume. (π ≈ 3.14)
Answer
282.6 m³
V = πr²h = 3.14 × 3² × 10 = 3.14 × 9 × 10 = 3.14 × 90 = 282.6 m³.
Can a triangle be formed with sides 5 cm, 7 cm, and 13 cm? Explain.
Answer
No
By the Triangle Inequality, the sum of any two sides must be GREATER than the third. Check the two smaller sides: 5 + 7 = 12, which is NOT greater than 13. Therefore, these sides CANNOT form a triangle.
A 1.8 m tall teacher casts a 2 m shadow. At the same time, a tree casts a 15 m shadow. How tall is the tree?
Answer
13.5 m
Using similar triangles: height/shadow ratio is constant. 1.8/2 = h/15. Cross-multiply: 2h = 1.8 × 15 = 27. h = 27 ÷ 2 = 13.5 m.
A cone has a radius of 6 cm and a vertical height of 8 cm. Find its volume. (π ≈ 3.14)
Answer
301.44 cm³
V = (1/3)πr²h = (1/3)(3.14)(6²)(8) = (1/3)(3.14)(36)(8) = (1/3)(3.14)(288) = (1/3)(904.32) = 301.44 cm³.
Find the distance between points A(−1, 2) and B(3, 5) on the coordinate plane.
Answer
5 units
d = √[(3−(−1))² + (5−2)²] = √[4² + 3²] = √[16 + 9] = √25 = 5 units. This is a 3-4-5 Pythagorean triple.
A trapezoidal lot has parallel sides of 14 m and 8 m and a perpendicular height of 5 m. Find its area.
Answer
55 m²
Area of trapezoid = ½ × (a + b) × h = ½ × (14 + 8) × 5 = ½ × 22 × 5 = ½ × 110 = 55 m².
Each interior angle of a regular polygon measures 144°. How many sides does the polygon have?
Answer
10 sides (Decagon)
Each interior angle = (n−2)×180°/n = 144°. So (n−2)×180 = 144n → 180n − 360 = 144n → 36n = 360 → n = 10.
A sphere has a radius of 6 cm. Find its surface area. (π ≈ 3.14)
Answer
452.16 cm²
SA = 4πr² = 4 × 3.14 × 6² = 4 × 3.14 × 36 = 4 × 113.04 = 452.16 cm².
Diagrams
Plane and Solid Geometry – Complete Concept Overview
A comprehensive mind map showing all major topics in Plane and Solid Geometry organized hierarchically from the central concept outward: angles and angle pairs, triangle classification and theorems, the Pythagorean theorem, polygons and their angle formulas, circle parts and formulas, plane area formulas for all figures, and solid geometry volume and surface area formulas.
Which Geometry Formula Do I Use? – LET Problem-Solving Decision Tree
A branching decision-tree flowchart that guides LET examinees from reading a geometry problem to selecting the correct formula. The first decision is WHAT the problem asks for (perimeter, area, volume, or surface area), which leads to WHICH FIGURE is involved, and then the specific formula to apply. Final nodes remind students of the correct unit type (linear, square, or cubic).
Step-by-Step: Solving Pythagorean Theorem Problems
A step-by-step flowchart for solving the three types of Pythagorean theorem problems on the LET: finding the hypotenuse (add squares, take root), finding a missing leg (subtract squares, take root), and testing whether three given sides form a right triangle (compare sum of smaller squares to square of largest side). Ends with a reminder of the key Pythagorean triples to memorize.
Triangle Congruence and Similarity – Which Criterion Applies?
A decision flowchart for choosing between congruence and similarity criteria for triangles. The first branch separates congruence (same size and shape) from similarity (same shape, proportional sizes). Each branch then lists all valid criteria with their conditions, helping LET examinees quickly identify which theorem to cite. The similarity branch reminds students that sides are in ratio k while areas scale by k².
Distribution of Geometry Sub-topics on LET Mathematics
A pie chart showing the approximate relative frequency of each geometry sub-topic in LET Mathematics items, based on past exam patterns. Area and perimeter of plane figures and volume/surface area of solids together account for over half of geometry items, making these the highest-priority topics for LET review. The Pythagorean theorem is the third most common, followed by angle and polygon problems, with congruence/similarity appearing least frequently.
Exam strategy
On the LET Mathematics component, Plane and Solid Geometry items reward careful formula recall and disciplined unit handling. Follow this approach for every geometry item: (1) READ carefully — identify whether the problem asks for length/perimeter (linear units), area (square units), or volume (cubic units); choosing the wrong formula type is the top error. (2) IDENTIFY the figure — is it a triangle, circle, cylinder, cone, etc.? Write the correct formula before substituting any values. (3) CHECK UNITS — if measurements are in different units (e.g., cm and m), convert BEFORE computing. (4) USE SHORTCUTS — recognize Pythagorean triples (3-4-5, 5-12-13, 8-15-17) to avoid square-root computation; use π = 22/7 when r is a multiple of 7. (5) REMEMBER THE ONE-THIRD RULE — cones and pyramids are always 1/3 of the matching prism or cylinder. (6) FOR SIMILAR FIGURES — sides are in ratio k, but areas are in ratio k². (7) VERIFY with context — does your answer make sense? A volume cannot be negative; an area in m² for a room should be roughly 20–60 m². Allocate about 1.5–2 minutes per geometry item; if stuck, eliminate obviously wrong units and use estimation. In Philippine elementary classrooms (K–12), geometry is taught from Grade 1 onward — knowing these concepts deeply also prepares you to teach them effectively, which is the ultimate goal of passing the LET under RA 7836.