Home / Courses / Geometry / Unit 6

Geometry Unit 6: Area of polygons & circles

Area measures the space inside a 2D shape in square units. Most formulas come from the rectangle.

Covers: Rectangles, triangles, parallelograms, trapezoids, regular polygons, circles and sectors

Study guide

Area measures the surface inside a figure, in square units. The basic formulas: rectangle A = lw; triangle A = ½bh, where the height is perpendicular to the base; parallelogram A = bh; trapezoid A = ½(b₁ + b₂)h, the average of the parallel bases times the height. A rhombus or kite can also use A = ½d₁d₂ with its diagonals. Perimeter is the distance around and is measured in plain units.

A regular polygon's area is A = ½·apothem·perimeter, where the apothem is the distance from the center to the midpoint of a side. You can see why by splitting the polygon into congruent triangles from the center. Finding the apothem often uses special right triangles or trigonometry.

For circles, the area is πr² and the circumference is 2πr (or πd). A sector is a pie-shaped slice; its area is (θ/360°)·πr², the fraction of the circle times the full area.

Composite figures are made of simpler shapes. Add the areas of the pieces, or subtract a cut-out (for example, the shaded region between a square and an inscribed circle). When a figure is drawn on a coordinate grid, find lengths with the distance formula or by counting.

If you know two sides a and b and the included angle C, the triangle's area is ½ab·sin C. This works for any triangle, not only right triangles. If the height isn't given, drop a perpendicular to the base and use a trig ratio (for example, h = side·sin of the angle) to find it before using ½bh.

Key ideas

• Rectangle: lw. Parallelogram: bh. Triangle: ½bh.
• Trapezoid: average the bases, times height.
• Circle: πr². Circumference: 2πr.
• Sector: a fraction of the circle, (θ/360)·πr².
• Composite shapes: add or subtract simpler pieces.

Formulas and rules

Common mistakes

• Using a slanted side instead of the height.
• Using diameter instead of radius.

Words to know

Area of a trapezoid

½(b₁ + b₂)h

Area of a circle

πr²

Area of a sector

(θ/360)·πr²

Area of a parallelogram

base × height

Units of area

Square units.

Area of a rectangle

lw

Area of a triangle

½bh, with the height perpendicular to the base.

Apothem

The distance from a regular polygon's center to the midpoint of a side.

Area of a regular polygon

½ × apothem × perimeter

Area of a rhombus or kite

½d₁d₂ (half the product of the diagonals)

Circumference

2πr (or πd)

Composite figure

A shape made of simpler shapes; add or subtract their areas.

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.