Volume formulas for the four most common 3D solids
Rectangular prism: l×w×h. Cylinder: πr²h (circle area × height). Sphere: 4/3×πr³. Cone: 1/3×πr²h (one-third of cylinder). Pyramid: 1/3 × base area × height.
Polygon Angle Sum
Interior angle sum = (n-2) × 180°
Polygon Angle Sum
Formula for the sum of interior angles of any polygon
Triangle (n=3): 180°. Quadrilateral (n=4): 360°. Pentagon (n=5): 540°. Hexagon (n=6): 720°. Each additional side adds 180°. Divide by n for each angle of a regular polygon.
Surface Area Formulas
Surface area of a sphere: SA (Surface Area) = 4 pi r squared. Cylinder: SA = 2πr² + 2πrh.
Surface Area Formulas
Surface area formulas for the most common 3D solids
Sphere: 4πr² (four times the area of a great circle). Cylinder: 2πr² (two circles) + 2πrh (the curved side unrolled into a rectangle). Cone: πr² + πrl where l = slant height.
Every square is a rectangle, but not every rectangle is a square
Parallelogram: two pairs of parallel sides. Rectangle: parallelogram with right angles. Rhombus: parallelogram with equal sides. Square: both rectangle and rhombus. Trapezoid (US): exactly one pair of parallel sides. Kite: two pairs of consecutive equal sides. Properties inherit down the hierarchy.
Diagonal Properties of Quadrilaterals
Diagonal properties: rectangle diagonals are equal. Rhombus diagonals are perpendicular bisectors of each other.
Diagonal Properties of Quadrilaterals
Key diagonal relationships for each special quadrilateral
Rectangle: diagonals are equal length and bisect each other. Rhombus: diagonals are perpendicular and bisect each other (but not necessarily equal). Square: diagonals are equal, perpendicular, and bisect each other. Parallelogram: diagonals bisect each other. Kite: one diagonal is the perpendicular bisector of the other.
Regular Polygons
Regular polygon: all sides equal AND all angles equal. Interior angle = (n-2)×180°/n
Regular Polygons
Polygons with both equal sides and equal angles
Regular triangle (equilateral): 60° each. Regular quadrilateral (square): 90° each. Regular pentagon: 108°. Regular hexagon: 120°. Regular octagon: 135°. Formula: each interior angle = (n-2)×180°/n. Sum of exterior angles of ANY polygon = always 360°.
Prisms and Pyramids
Prism: two parallel congruent bases + rectangular sides. V = base area × height.
Prisms and Pyramids
Two families of 3D solids and their formulas
Prism: two congruent parallel polygonal bases connected by rectangles. V = B×h (B = base area). Lateral surface area = perimeter of base × height. Pyramid: one polygonal base, triangular sides meeting at apex. V = ⅓B×h. Cone: circular pyramid. Cylinder: circular prism.
Prism
Two parallel bases, V = B×h
Pyramid
One base, apex, V = ⅓B×h
Cylinder
Circular prism, V = πr²h
Cone
Circular pyramid, V = ⅓πr²h
Euler's Formula
Euler's formula for polyhedra: V - E + F = 2 (vertices minus edges plus faces = 2)
Euler's Formula
A remarkable relationship between vertices, edges, and faces
For any convex polyhedron: V - E + F = 2. Cube: 8 vertices - 12 edges + 6 faces = 2 ✓. Tetrahedron: 4 - 6 + 4 = 2 ✓. Octahedron: 6 - 12 + 8 = 2 ✓. Euler characteristic. Used in topology — generalizes to non-convex and non-simply-connected surfaces.
Similar Figures
Similar figures: all corresponding angles equal, all corresponding sides proportional. Scale factor k → area scales k².
Similar Figures
Proportional shapes — and how area and volume scale
Similar: same shape, different size. Ratio of corresponding sides = scale factor k. Area ratio = k². Volume ratio = k³. If scale factor is 2: area is 4× larger, volume is 8× larger. Useful for: maps, scale models, indirect measurement.
Surface Area Distinctions
Lateral vs total surface area: lateral = sides only. Total = lateral + base(s).
Surface Area Distinctions
Understanding which surfaces to include in area calculations
Lateral surface area: only the sides, not the top or bottom. Total surface area: all surfaces including bases. Cylinder lateral SA = 2πrh. Cylinder total SA = 2πrh + 2πr². Cone lateral SA = πrl (l = slant height). Cone total SA = πrl + πr². Unroll the surface mentally to find the shape to calculate.
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🎓 Common Exam Questions
Q: What are the volume formulas for all major 3D shapes and how do they relate?
A: Prism (and cylinder): V = base area times height. The base can be any polygon — rectangular prism is length times width times height, cylinder is pi r squared times h. Pyramid (and cone): V = (1/3) times base area times height — exactly one third of the prism with the same base and height. Sphere: V = (4/3) pi r cubed. Key relationships: cone = 1/3 cylinder. Pyramid = 1/3 prism. This 1/3 relationship can be demonstrated by filling a cone-shaped container and pouring into a cylinder three times to fill it.
Q: Explain the polygon angle sum formula (n-2) times 180 — why does it work?
A: For any convex polygon with n sides: interior angle sum = (n-2) times 180 degrees. Why: any polygon can be divided into triangles by drawing diagonals from one vertex. A polygon with n sides creates n-2 triangles. Each triangle has 180 degrees of interior angles. So total = (n-2) times 180. Examples: triangle (3 sides): (3-2) times 180 = 180 degrees. Quadrilateral: (4-2) times 180 = 360 degrees. Pentagon: 540 degrees. Hexagon: 720 degrees. Regular polygon: each interior angle = (n-2) times 180 / n. Regular hexagon: 720/6 = 120 degrees each.
Q: Explain similar figures — what ratios apply to sides, areas, and volumes?
A: Similar figures have the same shape but different size — corresponding angles are equal and corresponding sides are proportional. If the ratio of sides (scale factor) is k: ratio of perimeters = k (linear — same as sides). Ratio of areas = k squared (since area involves two dimensions). Ratio of volumes = k cubed (since volume involves three dimensions). Example: two similar cylinders have radii 2 and 6, so k = 3. Volume ratio = 27:1 — the larger cylinder holds 27 times more. This is why doubling the size of a container increases its capacity eightfold (2 cubed = 8).
Q: What is Euler's formula V - E + F = 2 and why does it hold for polyhedra?
A: Euler's formula: for any convex polyhedron, vertices minus edges plus faces = 2. Examples: cube (V=8, E=12, F=6): 8-12+6=2. Tetrahedron (V=4, E=6, F=4): 4-6+4=2. Octahedron (V=6, E=12, F=8): 6-12+8=2. Why it holds: the formula relates to the topology (connectivity) of the polyhedron. It can be proven by projecting the polyhedron onto a plane (flattening it into a planar graph) and then systematically removing edges — each step preserves V-E+F. The formula fails for shapes with holes (like a torus) where V-E+F=0. It is a fundamental result connecting geometry and topology.
Q: Explain the quadrilateral hierarchy — what properties does each shape inherit?
A: Hierarchy from most general to most specific: Quadrilateral (4 sides). Trapezoid (at least one pair of parallel sides). Parallelogram (two pairs of parallel sides — inherits all trapezoid properties plus: opposite sides equal, opposite angles equal, diagonals bisect each other). Rectangle (parallelogram with all right angles — diagonals are also equal). Rhombus (parallelogram with all sides equal — diagonals are perpendicular and bisect the angles). Square (rectangle AND rhombus — all sides equal AND all right angles — diagonals are equal, perpendicular, bisect each other and the angles). Key: every square is a rectangle, every rectangle is a parallelogram, but not vice versa.