LET Elementary Mathematics — Plane and Solid GeometrySummary

An angle is a measure of rotation, expressed in degrees. Acute angles measure less than 90°, right angles measure exactly 90°, obtuse angles measure between 90° and 180°, straight angles measure exactly 180°, and reflex angles measure between 180° and 360°. Understanding angle measures is the gateway to all geometric reasoning. In the classroom, you will use a protractor to teach pupils how to measure and draw angles; place the protractor's center on the vertex and read the degree mark where the second ray falls. Key angle pairs include complementary angles (sum to 90°), supplementary angles (sum to 180°), and vertical angles (opposite angles formed by two intersecting lines, always equal). When a transversal crosses two parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles are supplementary. Mastering these relationships unlocks multi-step reasoning items on the LET.

Triangles are among the most important shapes in geometry and appear in multiple forms on the LET. The interior angles of any triangle always sum to 180°, a rule that persists regardless of the triangle's shape or size. Triangles are classified by sides: equilateral triangles have all three sides equal (and all angles 60°), isosceles triangles have two equal sides (and two equal base angles), and scalene triangles have no equal sides. They are also classified by angles: acute triangles have all angles less than 90°, right triangles have one 90° angle, and obtuse triangles have one angle greater than 90°. An exterior angle of a triangle equals the sum of the two remote interior angles, a property that often yields quick solutions to test items. The triangle inequality theorem states that the sum of any two sides must exceed the third side; if this fails, the three lengths cannot form a triangle. For example, sides 3, 4, and 8 cannot form a triangle because 3 + 4 = 7, which is not greater than 8. Congruence is established by five criteria: SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), AAS (two angles and a non-included side equal), and HL (hypotenuse and one leg of a right triangle equal). Similarity (same shape, proportional sides) is proved by AA (two angles equal), SSS-proportional (all sides proportional), or SAS-proportional (two sides proportional with the included angle equal). In similar triangles, corresponding angles are equal and corresponding sides are in the same ratio; crucially, areas scale by the square of that ratio.

In any right triangle with legs a and b and hypotenuse c, the relationship a² + b² = c² holds. This is one of the most powerful tools in geometry and appears repeatedly on the LET. To apply it: identify the right angle (the angle marked with a square); label the two sides forming the right angle as a and b, and the side opposite the right angle as c (the hypotenuse, the longest side); then plug into the formula. For example, if a ladder leans against a wall with its foot 6 m from the wall and reaches 8 m up the wall, the ladder length is c = √(6² + 8²) = √(36 + 64) = √100 = 10 m. The theorem also identifies right triangles: if a² + b² = c² for a given set of side lengths, the triangle is right-angled. Common Pythagorean triples—sets of integers satisfying a² + b² = c²—are 3-4-5, 5-12-13, and 8-15-17, plus all their multiples (6-8-10, 9-12-15, 10-24-26). Recognizing these triples allows you to skip the square root calculation and answer test items faster. The theorem also extends to distances on the coordinate plane: the distance between points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²], which is the Pythagorean theorem applied to horizontal and vertical displacements.

A polygon is a closed, two-dimensional figure made of straight sides. For any polygon with n sides, the sum of interior angles is (n − 2) × 180°. For example, a triangle (n = 3) has angle sum (3 − 2) × 180° = 180°; a quadrilateral (n = 4) has sum (4 − 2) × 180° = 360°; a hexagon (n = 6) has sum (6 − 2) × 180° = 720°. In a regular polygon (all sides equal, all angles equal), each interior angle measures [(n − 2) × 180°] ÷ n. So each interior angle of a regular hexagon is 720° ÷ 6 = 120°. The sum of exterior angles of any convex polygon is always 360°, regardless of n; this is a quick check on your work. Quadrilaterals (four-sided polygons) form a family with important inclusions: every square is both a rectangle and a rhombus; every rectangle and rhombus is a parallelogram. A square has four equal sides and four right angles. A rectangle has opposite sides equal and four right angles. A rhombus has four equal sides and opposite angles equal. A parallelogram has opposite sides parallel and equal. A trapezoid has exactly one pair of parallel sides. Understanding these hierarchies helps you recognize which properties apply; for instance, if a quadrilateral is a square, then it must also be a rectangle, so you can immediately say all its angles are 90°.

Perimeter is the total length around a figure, measured in linear units (cm, m, km). Area is the space enclosed by a figure, measured in square units (cm², m², km²). Always ensure all lengths are in the same unit before computing, or your answer will be off by a factor of 10, 100, or 10,000. For a square with side s, perimeter is 4s and area is s². For a rectangle with length l and width w, perimeter is 2(l + w) and area is l × w. For a triangle with base b and height h, area is ½ × b × h; perimeter is the sum of all three sides. Note that height is the perpendicular distance from the base to the opposite vertex, not the length of a slanted side. For a parallelogram with base b and height h, area is b × h; again, height is perpendicular. For a trapezoid with parallel sides a and b and height h, area is ½(a + b) × h. For a circle with radius r, circumference (the perimeter) is C = 2πr or C = πd (where d = 2r is the diameter), and area is πr². Use π ≈ 3.1416 in most problems, but when the radius is a multiple of 7 (e.g., 7, 14, 21), use π = 22/7 because it simplifies the fraction elegantly: a circle with r = 7 has area (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 cm² without decimals. For composite figures (like an L-shaped lot or a rectangle topped by a semicircle), decompose the figure into standard shapes, calculate the area of each, and add them. Alternatively, if a shape is a larger figure minus a cut-out, calculate the area of the whole, subtract the cut-out area, and verify your result.

A circle is the set of all points equidistant from a fixed center point. The radius (r) is the distance from the center to any point on the circle; the diameter (d) is the distance across the circle through the center (d = 2r). A chord is any line segment joining two points on the circle; a diameter is a special chord through the center. An arc is a portion of the circumference (the perimeter of the circle). A central angle has its vertex at the center of the circle. A tangent is a line that touches the circle at exactly one point and is perpendicular to the radius at that point—a key property for real-world applications like roads and wheels. The circumference C = πd = 2πr relates the perimeter to the radius; the area A = πr² relates the enclosed space to the radius. Because π is the ratio of every circle's circumference to its diameter, knowing any one of radius, diameter, circumference, or area lets you find the rest. For example, if a circular table has a circumference of 44 cm and you use π = 22/7, then 2πr = 44, so 2 × (22/7) × r = 44, yielding r = 7 cm, and the area is πr² = (22/7) × 49 = 154 cm².

While plane geometry deals with two-dimensional shapes, solid geometry involves three-dimensional objects. Volume is the space occupied by a solid, measured in cubic units (cm³, m³). Surface area is the total area of all the faces and curved surfaces of a solid, measured in square units (cm², m²). For a cube with edge length s, volume is V = s³ and surface area is SA = 6s² (six square faces). For a rectangular prism (or box) with length l, width w, and height h, volume is V = l × w × h and surface area is SA = 2(lw + lh + wh) (the sum of the areas of six rectangular faces, counted in pairs). For a cylinder with radius r and height h, volume is V = πr²h (area of the base times height) and surface area is SA = 2πr² + 2πrh (two circular bases plus the lateral surface). For a cone with radius r, height h, and slant height l (the distance along the sloped surface from the apex to the base circle), volume is V = (1/3)πr²h and surface area is SA = πr² + πrl (the base plus the lateral surface). For a sphere with radius r, volume is V = (4/3)πr³ and surface area is SA = 4πr². For a pyramid with base area B and height h, volume is V = (1/3)Bh. A crucial recurring insight: the volume of a cone or pyramid is exactly one-third the volume of a cylinder or prism with the same base and height. This relationship helps you spot wrong answers and verify your calculations. Always work in the same unit system; converting between cm³, mm³, and m³ requires dividing or multiplying by 1000, and mistakes here are common.

Two figures are congruent if they have the same shape and size; one can be moved (translated, rotated, reflected) to fit exactly onto the other. Congruent triangles have corresponding sides equal and corresponding angles equal. The five congruence tests for triangles are SSS (side-side-side: all three pairs of sides equal), SAS (side-angle-side: two sides and the included angle equal), ASA (angle-side-angle: two angles and the included side equal), AAS (angle-angle-side: two angles and a non-included side equal), and HL (hypotenuse-leg: the hypotenuse and one leg of a right triangle equal). You do not need to show all six elements are equal; meeting the criteria for one of these five tests guarantees full congruence. Two figures are similar if they have the same shape but not necessarily the same size; one is an enlargement or reduction of the other. Similar triangles have corresponding angles equal and corresponding sides in the same ratio (proportional). The tests for similarity are AA (two angles equal, which forces the third angle equal), SSS-proportional (all three pairs of sides in the same ratio), and SAS-proportional (two sides in the same ratio with the included angle equal). A critical insight: if two figures are similar with a side-length ratio of k, then their areas have a ratio of k², and if they are three-dimensional, their volumes have a ratio of k³. For example, if one triangle has sides twice as long as another similar triangle, the first has an area four times larger; if one cube has edges three times longer, it has a volume 27 times larger. This scaling relationship unlocks many LET problems.

A critical discipline in geometry is ensuring all measurements are in the same unit before performing calculations. If a rectangle has length 2 m and width 50 cm, you must convert one to match the other: either 2 m = 200 cm (so area = 200 × 50 = 10,000 cm²) or 50 cm = 0.5 m (so area = 2 × 0.5 = 1 m² = 10,000 cm²). Mixing units is one of the most common errors on the LET. When solving a geometry problem: (1) read the question carefully to identify which measurement is requested (perimeter, area, surface area, or volume—each uses a different formula); (2) identify the shape and its given dimensions; (3) ensure all dimensions are in the same unit; (4) select the appropriate formula and substitute the values; (5) perform the calculation step-by-step, showing your work; (6) verify that your answer has the correct unit (linear for perimeter, square for area, cubic for volume); and (7) check reasonableness (e.g., the area of a shape should be smaller than a much larger surrounding shape). This systematic approach minimizes errors and builds confidence.

Placing figures on the Cartesian plane (with an x-axis and y-axis) allows you to calculate lengths and positions using formulas grounded in the Pythagorean theorem. The distance formula gives the distance between two points (x₁, y₁) and (x₂, y₂) as d = √[(x₂ − x₁)² + (y₂ − y₁)²]. For instance, the distance between (1, 2) and (4, 6) is d = √[(4 − 1)² + (6 − 2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5 units. The midpoint formula locates the center of a segment joining two endpoints: midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). For example, the midpoint of the segment from (2, 3) to (8, 7) is ((2 + 8)/2, (3 + 7)/2) = (5, 5). These formulas appear in LET items about segment length, shape position, and area on the coordinate plane.

The fact that interior angles of a triangle sum to 180° is a direct application of the angle-sum principle. If you know two angles, you immediately calculate the third. This relationship extends to all polygons via the formula (n − 2) × 180°.

The Pythagorean theorem applies exclusively to right triangles (one 90° angle). Identifying a right angle signals that you can use a² + b² = c². Conversely, if a² + b² = c² holds, the triangle must be right-angled.

When two figures are similar with a side-length ratio of k, their areas are in a ratio of k². This scaling relationship is critical for shadow problems, map interpretations, and scale-model questions on the LET.

Triangles are the building blocks of polygons. A polygon can be decomposed into triangles, so understanding triangle angle sums unlocks the angle-sum formula for any polygon: (n − 2) × 180°.

Both circumference (C = 2πr) and area (A = πr²) depend on the radius r, linked by the constant π. Knowing one measurement (e.g., radius or circumference), you can derive all others.

The concept of perpendicular height is central to the area of triangles (½ × b × h), parallelograms (b × h), and trapezoids (½(a + b) × h). Misidentifying height (confusing it with a slant side) is a common error.

Volume formulas for prisms and cylinders extend the area-of-the-base formula by multiplying by height (V = base area × h). Cones and pyramids scale this by one-third (V = ⅓ × base area × h).

The distance formula on the coordinate plane, d = √[(x₂ − x₁)² + (y₂ − y₁)²], is a direct application of the Pythagorean theorem to horizontal and vertical displacements.

When a transversal crosses two lines, the angle relationships (corresponding angles equal, alternate interior angles equal, co-interior angles supplementary) determine whether the two lines are parallel. This is tested in geometric reasoning on the LET.

All formulas for area, volume, perimeter, and circumference assume consistent units. Converting units mid-calculation is a common source of error; always standardize units first.