Math — Geometry

Basic Geometry Shapes

From triangles to cylinders, learn the names, properties, and formulas for the most important shapes in mathematics.

Learning Objectives

  • Name and classify the major 2D shapes and their properties
  • Distinguish between types of triangles and quadrilaterals
  • Identify common 3D (solid) shapes and describe their faces, edges, vertices
  • Apply area formulas for rectangles, triangles, and circles
  • Understand angle sums for polygons

Prerequisites

Basic multiplication and division. Familiarity with the concept of angles (knowing what 90° looks like is enough). Khan Academy Geometry has free warm-up exercises if needed.

The Lesson

Step 1 — 2D Shapes: Triangles

A triangle has exactly 3 sides and 3 angles. The angles always sum to 180°. Triangles are classified two ways: by side length and by angle.

Equilateral
3 equal sides · 3 × 60° angles
Isosceles
2 equal sides · 2 equal base angles
Scalene
No equal sides · No equal angles
Right Triangle
One 90° angle · Pythagorean theorem applies
Area of a triangle: A = ½ × base × height
Example: base = 8 cm, height = 5 cm → A = ½ × 8 × 5 = 20 cm²

Step 2 — Quadrilaterals

Any 4-sided polygon. Angle sum = 360°. The key quadrilaterals form a hierarchy:

Square
4 equal sides · 4 × 90° · diagonals bisect perpendicularly
Rectangle
Opposite sides equal · 4 × 90°
Rhombus
4 equal sides · opposite angles equal
Parallelogram
Opposite sides parallel and equal
Trapezoid
Exactly one pair of parallel sides
Kite
Two pairs of adjacent equal sides
Area of a rectangle: A = length × width. Example: 9 m × 4 m = 36 m²
Area of a parallelogram: A = base × height. Example: base 6, height 4 = 24 units²

Step 3 — Circles

A circle is defined by its radius (distance from centre to edge) and diameter (twice the radius). The key constant is π ≈ 3.14159.

Circumference (perimeter): C = 2πr  |  Example: r = 7 cm → C = 2 × 3.14159 × 7 ≈ 43.98 cm
Area: A = πr²  |  Example: r = 5 cm → A = 3.14159 × 25 ≈ 78.54 cm²

Step 4 — Other Polygons

The interior angle sum of any polygon = (n − 2) × 180° where n = number of sides.

Pentagon (n=5): (5−2) × 180° = 3 × 180° = 540°
Hexagon (n=6): (6−2) × 180° = 4 × 180° = 720°

Step 5 — 3D Shapes

Three-dimensional shapes have faces (flat surfaces), edges (lines where faces meet), and vertices (corners). Euler's formula for convex polyhedra: Faces + Vertices − Edges = 2.

Cube
6 faces · 12 edges · 8 vertices · Volume = s³
Rectangular Prism
6 faces · Volume = l × w × h
Cylinder
2 circular faces + curved surface · V = πr²h
Sphere
0 flat faces · V = (4/3)πr³
Cone
1 circular base · 1 apex · V = (1/3)πr²h
Pyramid
Polygon base + triangular faces meeting at apex

Practice Problems

  1. Q: A triangle has angles 45° and 70°. What is the third angle?
    Solution: 180° − 45° − 70° = 65°
  2. Q: What is the area of a circle with diameter 10 cm?
    Solution: r = 5 cm → A = π × 5² = π × 25 ≈ 78.54 cm²
  3. Q: A rectangular prism is 4 cm × 3 cm × 6 cm. What is its volume?
    Solution: V = 4 × 3 × 6 = 72 cm³
  4. Q: What is the interior angle sum of an octagon (8 sides)?
    Solution: (8−2) × 180° = 6 × 180° = 1080°
  5. Q: A cube has a side length of 3 m. What is its surface area?
    Solution: Surface area = 6 × s² = 6 × 9 = 54 m²

Common Mistakes

Mistake 1 — Using diameter instead of radius in circle formulas. If you're given diameter = 12, remember r = 6. A = πr² = π(6)² = 36π, not π(12)².
Mistake 2 — Forgetting that all squares are rectangles (but not vice versa). A square satisfies every property of a rectangle and a rhombus simultaneously.
Mistake 3 — Adding angles of a triangle to 360° instead of 180°. Quadrilaterals sum to 360°; triangles sum to 180°.
Mistake 4 — Confusing perimeter with area. Perimeter is the total distance around a shape (1D measurement in cm, m). Area is the space inside (2D in cm², m²).
Mistake 5 — Mixing up 2D and 3D formulas. A cylinder's volume needs πr²h; its surface area needs 2πr² + 2πrh. Keep them separate.

Further Practice Resources

Khan Academy — Geometry (comprehensive course with exercises) Math Is Fun — Geometry Index (interactive shape explorer) Britannica — Geometry overview and history Wikipedia — List of 2D Geometric Shapes MIT OCW — Advanced mathematics foundations

Frequently Asked Questions

What are the main types of triangles?

Equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal), right (one 90° angle), acute (all angles less than 90°), obtuse (one angle greater than 90°).

How many sides does a polygon have?

A polygon can have any number of sides from 3 upward: triangle=3, quadrilateral=4, pentagon=5, hexagon=6, heptagon=7, octagon=8, and so on.

What is the difference between a square and a rectangle?

A square is a special rectangle where all four sides are equal. Every square is a rectangle, but not every rectangle is a square.

What is the formula for the area of a circle?

Area = π × r² where r is the radius. π ≈ 3.14159.

What makes a shape 3D?

A 3D shape has length, width, AND depth — it occupies volume. A flat 2D shape only has length and width.