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Grades 8-10Geometry

Geometry Foundations

Foundations of geometry: points, lines, angles and their relationships; triangle classification and the 180-degree angle sum; quadrilaterals and polygons; perimeter and circumference; area of 2D shapes; and volume of prisms and cylinders. Learn each concept, then practice with instant feedback.

1Points, Lines, and Angles

Key idea: A point marks a location, lines and rays are built from points, and an angle is two rays sharing a vertex — measured in degrees.
  1. A point is a single location, named with a capital letter (point A).
  2. A segment connects two endpoints; a ray starts at one endpoint and goes forever; a line goes forever in both directions.
  3. An angle is two rays with a common endpoint (the vertex), measured in degrees.
  4. Classify angles by size: acute (under 90°), right (exactly 90°), obtuse (over 90°).
Worked example
An angle measures 130°. Is it acute, right, or obtuse?
130° is over 90°, so it is obtuse.
Worked example
Which has exactly one endpoint: a segment, a ray, or a line?
A ray: it starts at its endpoint and goes on forever in one direction. A segment has two endpoints; a line has none.
Watch out: A right angle is EXACTLY 90°. An 89° angle is acute, not 'basically right' — precision matters in geometry!

2Angle Relationships

Key idea: Pairs of angles follow strict sum rules: complementary angles add to 90°, supplementary angles add to 180°, and vertical angles are always equal.
  1. Complementary angles add to 90° — think C for Corner: a square corner is 90°.
  2. Supplementary angles add to 180° — think S for Straight line, which is 180°.
  3. When two lines cross, opposite (vertical) angles are always EQUAL.
  4. Adjacent angles on a straight line (a linear pair) add to 180°.
  5. When parallel lines are cut by a transversal, corresponding angles and alternate interior angles are equal.
Worked example
What is the complement of 52°?
Complement means 90°: 90° − 52° = 38°.
Worked example
Two lines cross and one angle is 110°. What is the angle opposite to it?
Vertical angles are equal, so the opposite angle is also 110°.
Watch out: Complementary is 90°, not 100°! The classic mix-up: using 180° for complements and 90° for supplements.

3Triangle Classification

Key idea: Classify every triangle twice: by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse).
  1. Equilateral: 3 equal sides (all angles 60°). Isosceles: 2 equal sides (base angles equal). Scalene: no equal sides.
  2. Count the equal sides — tick marks on a diagram show which sides match.
  3. By angles: look at the BIGGEST angle — under 90° is acute, exactly 90° is right, over 90° is obtuse.
  4. Triangle inequality: the two shorter sides must add to MORE than the longest side, or it cannot be a triangle.
Worked example
Classify a triangle with sides 5 cm, 5 cm, 8 cm.
Two sides are equal, so it is isosceles. Check: 5 + 5 = 10 > 8, so it is a valid triangle.
Worked example
Classify a triangle with angles 30°, 60°, 90°.
The biggest angle is exactly 90°, so it is a right triangle.
Watch out: Isosceles needs only TWO equal sides — do not confuse it with equilateral. And 'scalene' means NO equal sides at all.

4Triangle Angle Sum

Key idea: A triangle's three interior angles always add to 180° — find a missing angle by subtracting the other two from 180°.
  1. Add the two known angles, then subtract from 180° to find the missing angle.
  2. An exterior angle equals the sum of the two remote (non-adjacent) interior angles.
  3. In an isosceles triangle the two base angles are equal — subtract the vertex angle from 180°, then divide by 2.
  4. In an equilateral triangle every angle is 60° (because 180° / 3 = 60°).
Worked example
Two angles are 47° and 83°. Find the third.
180° − 47° − 83° = 50°.
Worked example
Remote interior angles are 38° and 71°. Find the exterior angle.
Exterior = sum of the two remote interiors = 38° + 71° = 109°.
Watch out: Always use INTERIOR angles in the 180° sum. If a problem gives you an exterior angle, subtract it from 180° first to get the interior one.

5Quadrilaterals and Polygons

Key idea: Quadrilaterals are classified by their sides and angles; any n-sided polygon has interior angles summing to (n − 2) × 180°, while exterior angles always sum to 360°.
  1. Parallelogram: both pairs of opposite sides are parallel — opposite angles are EQUAL, consecutive angles add to 180°.
  2. Rectangle: 4 right angles (diagonals equal). Rhombus: 4 equal sides (diagonals perpendicular). Square: both — 4 equal sides AND 4 right angles.
  3. Trapezoid: exactly one pair of parallel sides.
  4. Polygon interior sum: (n − 2) × 180°. Example: hexagon (n = 6) → 4 × 180° = 720°.
  5. Exterior angles always add to 360°; a regular n-gon has each exterior angle = 360° / n.
Worked example
Find the interior angle sum of an octagon.
n = 8, so (8 − 2) × 180° = 6 × 180° = 1080°.
Worked example
A parallelogram has one angle of 72°. Find the opposite and adjacent angles.
Opposite is equal: 72°. Adjacent adds to 180°: 180° − 72° = 108°.
Watch out: A square IS a rectangle and a rhombus — special cases, not rivals. And interiors grow with n, but exteriors always sum to 360°.

6Perimeter and Circumference

Key idea: Perimeter is the distance around any shape (add every side); a circle's perimeter has a special name — circumference, C = πd = 2πr.
  1. Perimeter: walk around the shape and add every side. Rectangle: P = 2(length + width).
  2. In a circle, the radius (center to edge) is HALF the diameter — d = 2r.
  3. Circumference: C = π × d, or C = 2πr. Leave answers in terms of π unless told otherwise.
  4. To find r or d from C, work backwards: divide C by π to get d, then halve for r.
Worked example
A rectangle is 8 cm by 5 cm. Find the perimeter.
P = 2(8 + 5) = 2 × 13 = 26 cm.
Worked example
A circle has radius 9 cm. Find the circumference in terms of π.
C = 2πr = 2π × 9 = 18π cm.
Watch out: Do not mix up 2πr (circumference) with πr² (area)! Circumference uses the radius ONCE; area SQUARES it.

7Area of 2D Shapes

Key idea: Every area formula multiplies two perpendicular lengths — rectangle lw, triangle (1/2)bh, trapezoid (1/2)(b1 + b2)h, circle πr² — and area always comes in square units.
  1. Rectangle: length × width. Parallelogram: base × height.
  2. Triangle: 12 × base × height — the height must hit the base at a right angle.
  3. Trapezoid: average the two bases, then multiply by height: 12(b1 + b2)h.
  4. Circle: π × r² — square the radius FIRST, then multiply by π.
  5. Arcs and sectors are FRACTIONS of a whole circle: arc = (θ/360) × circumference, sector = (θ/360) × area.
Worked example
A trapezoid has bases 9 cm and 15 cm and height 6 cm. Find its area.
12(9 + 15)(6) = 12(24)(6) = 12 × 6 = 72 cm².
Worked example
A sector has central angle 120° and radius 9 cm. Find its area in terms of π.
Circle area = π × 9² = 81π. Sector = 120360 × 81π = 13 × 81π = 27π cm².
Watch out: The height must be PERPENDICULAR to the base — never use a slanted side as the height! And area is square units (cm²), not just cm.

8Volume of Prisms and Cylinders

Key idea: Volume = (area of the base) × height — B × h for prisms and πr²h for cylinders — and volume always comes in cubic units.
  1. A prism has two identical flat bases: V = (base area) × height. Rectangular box: V = length × width × height.
  2. Cylinder: the base is a circle, so V = πr² × h.
  3. Pointy solids take one third: cone V = 13πr²h, pyramid V = 13Bh.
  4. Volume is measured in CUBIC units (cm³) — three dimensions multiplied.
Worked example
A rectangular box is 5 cm × 8 cm × 3 cm. Find its volume.
V = 5 × 8 × 3 = 120 cm³.
Worked example
A cylinder has radius 4 cm and height 7 cm. Find its volume in terms of π.
V = π × 4² × 7 = π × 16 × 7 = 112π cm³.
Watch out: Square the radius before multiplying by height! And the height used in the formula is perpendicular to the base — not a slant.
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