Plane shapes · 3 min read
Rectangle Area: the method, worked through
A rectangle partitions into equal rows and columns of square units.
30 illustrated guides
Shape studies
Work through the formula, compare a few scenarios and know where the model stops.
Plane shapes · 3 min read
Understanding rectangle perimeter
Perimeter measures the full boundary of a rectangle.
Plane shapes · 3 min read
Rectangle Diagonal: from inputs to a useful answer
The diagonal joins opposite corners and forms a right triangle with the two sides.
Plane shapes · 3 min read
A practical guide to square area
A square has two equal perpendicular dimensions.
Plane shapes · 3 min read
How square perimeter works
A square has four equal edges, so one side determines the full boundary.
Plane shapes · 3 min read
Square Diagonal: the method, worked through
A square diagonal is the hypotenuse of an isosceles right triangle.
Circles · 3 min read
Understanding circle area
Circle area depends on the square of the distance from the centre to the boundary.
Circles · 3 min read
Circle Circumference: from inputs to a useful answer
The circumference is the length around a circle.
Circles · 3 min read
A practical guide to circle radius from area
Inverting the circle-area relation recovers a radius from a known surface area.
Circles · 3 min read
How circle diameter from circumference works
The ratio of a circle's circumference to its diameter is constant, so division by pi reverses the measurement.
Triangles · 3 min read
Triangle Area: the method, worked through
A triangle fills half a parallelogram with the same base and perpendicular height.
Triangles · 3 min read
Understanding heron triangle area
Heron's formula finds triangle area from the three side lengths alone.
Triangles · 3 min read
Right Triangle Hypotenuse: from inputs to a useful answer
The hypotenuse sits opposite the right angle and is always the longest side of a right triangle.
Triangles · 3 min read
A practical guide to missing right triangle leg
Rearranging Pythagoras finds the missing perpendicular leg when the longest side and one leg are known.
Triangles · 3 min read
How equilateral triangle area works
An equilateral triangle divides into two matching right triangles, which explains the square-root-of-three factor.
Triangles · 3 min read
Equilateral Triangle Height: the method, worked through
The altitude of an equilateral triangle bisects its base and forms a thirty-sixty-ninety triangle.
Triangles · 3 min read
Understanding triangle third angle
Interior angles of a Euclidean triangle sum to one hundred eighty degrees.
Triangles · 3 min read
Law of Cosines Side: from inputs to a useful answer
The law of cosines extends the right-triangle relation to any included angle.
Plane shapes · 3 min read
A practical guide to parallelogram area
Sliding a corner triangle to the other side turns a parallelogram into a rectangle without changing its area.
Plane shapes · 3 min read
How trapezoid area works
A trapezoid's average parallel width multiplied by its perpendicular height gives the enclosed area.
Plane shapes · 3 min read
Rhombus Area: the method, worked through
The perpendicular diagonals of a rhombus split it into four right triangles.
Curves · 3 min read
Understanding ellipse area
An ellipse can be viewed as a circle stretched independently along two perpendicular axes.
Curves · 3 min read
Ellipse Perimeter Estimate: from inputs to a useful answer
Ramanujan's first perimeter approximation gives a convenient estimate without evaluating an elliptic integral.
Circles · 3 min read
A practical guide to sector area
A sector is a fraction of a circle selected by its central angle.
Circles · 3 min read
How arc length works
Arc length measures the curved boundary selected by a central angle.
Circles · 3 min read
Chord Length: the method, worked through
A chord is the straight segment between two points on a circle.
Circles · 3 min read
Understanding annulus area
An annulus is the ring left after removing a concentric smaller disk.
Polygons · 3 min read
Regular Polygon Area: from inputs to a useful answer
A regular polygon splits into congruent triangles around its centre.
Polygons · 3 min read
A practical guide to regular polygon interior angle
A regular polygon shares its total interior angle sum evenly across its vertices.
Polygons · 3 min read
How polygon diagonal count works
Each vertex connects by diagonals to all except itself and its two neighbours.
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