Basic Geometry Shapes and Their Properties

2026-05-16 · A2Z Lessons

Prerequisites

You should know what an angle is, that a straight line measures 180°, and that a full rotation measures 360°. Basic multiplication will help with area formulas. For a refresher on measurement, see our Area and Perimeter Calculations lesson.

The Language of Geometry

Geometry is the branch of mathematics concerned with shapes, sizes, and positions of figures. Before studying specific shapes, a few terms appear everywhere:

• Vertex (plural: vertices) — a corner point where two sides meet
• Side (or edge) — a straight line segment forming part of the shape's boundary
• Interior angle — the angle measured inside the shape at a vertex
• Diagonal — a line segment connecting two non-adjacent vertices
• Congruent — equal in size and shape
• Parallel — lines that run in the same direction and never meet (shown by arrows: →→)
• Perpendicular — meeting at a right angle (90°)

Triangles

A triangle has exactly 3 sides and 3 interior angles. The most important rule: the three interior angles of any triangle always sum to 180°. This is true whether the triangle is tiny or enormous, fat or thin.

Types of Triangle by Sides

• Equilateral — all 3 sides equal; all 3 angles = 60°
• Isosceles — exactly 2 sides equal; the 2 base angles are also equal
• Scalene — all 3 sides different lengths; all 3 angles different

Types of Triangle by Angles

• Acute — all angles less than 90°
• Right — one angle exactly 90°; the side opposite is called the hypotenuse
• Obtuse — one angle greater than 90°

Quadrilaterals

A quadrilateral has exactly 4 sides and 4 interior angles. Key rule: interior angles of any quadrilateral sum to 360°.

Common Quadrilaterals

• Square — 4 equal sides, 4 right angles (90° each), 2 lines of symmetry through diagonals + 2 through midpoints = 4 total
• Rectangle — opposite sides equal and parallel, 4 right angles, 2 lines of symmetry
• Rhombus — 4 equal sides, opposite angles equal, diagonals bisect each other at right angles
• Parallelogram — opposite sides equal and parallel, opposite angles equal, no right angles (unless it is a rectangle)
• Trapezoid (US) / Trapezium (UK) — exactly one pair of parallel sides
• Kite — two pairs of consecutive equal sides; one diagonal is a line of symmetry

Circles

A circle is not a polygon — it has no straight sides or vertices. Instead, every point on the circle is the same distance from the center. That distance is called the radius (r).

Key Circle Terms and Formulas

• Radius (r) — distance from center to any point on the circle
• Diameter (d) — distance straight across through the center; d = 2r
• Circumference (C) — distance around the circle; C = 2πr = πd
• Area (A) — space enclosed; A = πr²
• π (pi) ≈ 3.14159 (often approximated as 3.14 or 22/7)

Regular Polygons

A regular polygon has all sides equal AND all angles equal. The interior angle of a regular polygon with n sides is calculated with a simple formula.

Sum of Interior Angles for Any Polygon

The total of all interior angles in any polygon with n sides is (n − 2) × 180°. This works even for irregular polygons.

Practice Problems

1. A right triangle has one acute angle of 38°. Find the other acute angle.
   Solution: 180° − 90° − 38° = 52°.
2. Name two properties that make a square different from a generic rectangle.
   Solution: A square has (1) all four sides equal in length and (2) four lines of symmetry, whereas a rectangle only has two lines of symmetry and its adjacent sides need not be equal.
3. A circle has a diameter of 10 m. What is its area? (Use π ≈ 3.14)
   Solution: r = 10 ÷ 2 = 5 m. Area = 3.14 × 5² = 3.14 × 25 = 78.5 m².
4. A regular polygon has interior angles of 140°. How many sides does it have?
   Solution: 140 = (n−2)×180/n → 140n = 180n − 360 → 40n = 360 → n = 9 sides (a nonagon).
5. A parallelogram has two angles of 65° each. What are the other two angles?
   Solution: Opposite angles are equal, and adjacent angles are supplementary. So the other two angles each equal 180° − 65° = 115°.

Common Mistakes to Avoid

• Confusing radius and diameter. The radius is half the diameter. Using the full diameter in A = πr² doubles your answer. Always confirm which measurement the problem gives you.
• Assuming all parallelograms have right angles. A rectangle is a special parallelogram with right angles. A general parallelogram has oblique angles.
• Forgetting that the angle-sum rule changes per polygon. Triangles sum to 180°, quadrilaterals to 360°, pentagons to 540°, etc. Always compute (n−2) × 180° for a new polygon.
• Calling all four-sided figures squares. A square requires both equal sides AND right angles. A rhombus has equal sides but not necessarily right angles.

Frequently Asked Questions

How many degrees are in a triangle?

The interior angles of any triangle always add up to exactly 180°, regardless of size or shape.

What is the difference between a square and a rectangle?

Both have four right angles. A square has all four sides equal in length. A rectangle has two pairs of equal opposite sides, but adjacent sides are not necessarily equal.

What is the formula for the area of a circle?

Area = π × r², where r is the radius. A circle with radius 5 cm has area ≈ 3.14159 × 25 ≈ 78.54 cm².

How many sides does a polygon need to have?

A polygon is any closed flat shape with at least 3 straight sides. Triangles (3), quadrilaterals (4), pentagons (5), hexagons (6), and so on are all polygons.

What makes a shape a regular polygon?

A regular polygon has all sides equal in length AND all interior angles equal. A square is a regular quadrilateral; an equilateral triangle is a regular triangle.

Further Practice Resources

• Khan Academy — Geometry Course
• Math Is Fun — Geometry
• Wikipedia — Polygon
• Britannica — Geometry
• MIT OpenCourseWare — Mathematics Resources

Related Lessons

• Area and Perimeter Calculations
• Basic Physics Laws for Students
• Ratios and Proportions Explained
• Percentages: From Sales Discounts to Test Scores
• Decimal Arithmetic Step by Step