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Area of Similar Polygons is taught in the note for Area and Perimeter. Read it below and look for the section on this topic.

Area and Perimeter

12 min read

Perimeter and area of polygons

The perimeter is the distance round a shape and the area is the surface it covers.

Area formulasFive shapes with their area formulas: rectangle length times width, triangle half base times height, parallelogram base times height, trapezium half the sum of the parallel sides times the height, and circle pi times radius squared.lwRectangleA = l × wbasehTriangleA = ½ × b × hbasehParallelogramA = b × habhTrapeziumA = ½(a + b) × hrCircleA = π r²
ShapeAreaPerimeter
Rectanglel × w2(l + w)
Triangle½ × b × ha + b + c
Parallelogramb × h2(a + b)
Trapezium½ (a + b) hsum of sides
Rhombus½ d₁ d₂4a

For an irregular polygon, divide it into triangles and rectangles and add the areas.

Regular polygons

A regular polygon has equal sides and equal angles. With n sides of length s, it can be divided into n identical isosceles triangles meeting at the centre.

Angles in a polygonA regular pentagon with an exterior angle marked at each corner by extending one side. The five exterior angles of any polygon always add up to 360 degrees. Each exterior angle of this pentagon is 72 degrees and each interior angle is 108 degrees.Exterior anglesalways add up to 360°Regular pentagon:exterior = 360° ÷ 5 = 72°interior = 180° − 72° = 108°Interior sum = (n − 2) × 180°= 3 × 180° = 540°
n = number of sides, s = side length.
Worked exampleA regular hexagon

Find the area of a regular hexagon of side 10 cm.

  1. n = 6, s = 10.
  2. tan 30° = 0.5774.

Answer: About 259.8 cm².

Similar polygons

If two polygons are similar and their lengths are in the ratio k : 1, their areas are in the ratio k² : 1. Doubling the sides makes the area four times as big.

Lengths scaled by k change the area by k².
Try itRegular polygon

Choose the number of sides and the side length.

Check yourself The area of a triangle with base 10 and height 6 is
Check yourself If lengths are doubled the area is multiplied by
Check yourself The perimeter of a regular octagon with side 5 is

Key terms

Perimeter
The distance round a shape.
Area
The surface a shape covers.
Regular polygon
A polygon with equal sides and angles.
Scale factor
The ratio of corresponding lengths of similar shapes.

Summary

  • Learn the area formulas for the rectangle, triangle, parallelogram, trapezium and rhombus.
  • The area of a regular n-gon is ns²/(4 tan(180°/n)).
  • Similar figures: area ratio = (length ratio)².
  • Split irregular shapes into simple shapes.
Area of a Regular Polygon