AFCAT Numerical Ability · Area and Perimeter

Rhombus, Trapezium, Parallelogram and Regular Polygons

Every figure here is cut into triangles or measured from its diagonals: a rhombus by half the product of its diagonals, a trapezium by the mean of its parallel sides, a hexagon as six equilateral triangles.

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Why this matters

This is a formula page: half its questions are EASY and each one names its figure. The marks go on the rhombus, where the diagonals must be halved before Pythagoras gives the side, and on the parallelogram, where each height belongs to its own pair of sides.

Concept 1 of 4: Rhombus and kite: working from the diagonals

The diagonals of a rhombus cross at right angles and cut each other in half, so they split it into four equal right triangles. Those triangles give both the area and the side. A kite's diagonals also cross at right angles, so its area uses the same formula.

Definition

Area of a rhombus or kite $= \tfrac12 d_1 d_2$ .
Side of a rhombus $= \sqrt{\left(\tfrac{d_1}{2}\right)^2 + \left(\tfrac{d_2}{2}\right)^2}$ , and its perimeter is four sides.
A rhombus is also a parallelogram, so its area is side × height as well.
A square is a rhombus with equal diagonals: $\tfrac12 d^2$ , the formula from the rectangles page.

Rhombus

A = \tfrac12 d_1 d_2,\qquad \text{side} = \sqrt{\left(\tfrac{d_1}{2}\right)^2 + \left(\tfrac{d_2}{2}\right)^2}

Worked example

A rhombus has diagonals of 16 cm and 30 cm. Find its area and its perimeter.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Area and Perimeter · Quadrilaterals and Regular Polygons

The diagonals of a rhombus are 24 cm and 10 cm, respectively. The perimeter of the rhombus is

Halve the diagonals first

With diagonals 16 and 30 the side is

\sqrt{8^2 + 15^2} = 17

. Putting the whole diagonals into Pythagoras gives 34, twice the side.

Half the product

Diagonals 9 and 16 give an area of 72, not 144. The product of the diagonals is the rectangle drawn round the rhombus.

Drill 3 more on rhombus and kite: working from the diagonals

Concept 2 of 4: Trapezium, parallelogram and any quadrilateral

A parallelogram can be cut and slid into a rectangle with the same base and height. A trapezium is the average of its two parallel sides times the gap between them. Any other four-sided field is cut along a diagonal into two triangles.

Definition

Parallelogram: area = base × height. Each pair of sides has its own height, so $\text{side}_1 \times h_1 = \text{side}_2 \times h_2$ .
Trapezium: area $= \tfrac12(a + b) \times h$ , with $a, b$ the parallel sides and $h$ the distance between them.
Any quadrilateral with a diagonal $d$ and the perpendiculars $h_1, h_2$ dropped on it from the other two corners: area $= \tfrac12 d (h_1 + h_2)$ .
Given four sides and one angle: the diagonal opposite the known angle splits it into two triangles. Find each by the triangle formulas and add.

Trapezium and quadrilateral

A_{\text{trap}} = \tfrac12(a + b)h,\qquad A_{\text{quad}} = \tfrac12 d(h_1 + h_2)

Worked example

A parallelogram has sides of 20 cm and 16 cm. The distance between its longer sides is 8 cm. Find the distance between its shorter sides.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Area and Perimeter · Quadrilaterals and Regular Polygons

The adjacent sides of a parallelogram are 36 cm and 27 cm in length. If the distance between the shorter sides is 12 cm, the distance between the longer sides is

Each height goes with its own sides

Sides 20 and 16 with a height of 8 on the 20 cm sides give an area of 160. Multiplying 8 by 16 pairs the height with the wrong side and gives 128.

The slant side is not the height

In a trapezium the height is the perpendicular gap between the parallel sides. A slanting side of 10 cm tells you nothing until it is turned into that gap.

Drill 5 more on trapezium, parallelogram and any quadrilateral

Concept 3 of 4: Equilateral triangles and the regular hexagon

An equilateral triangle's height cuts it into two right triangles with legs

\tfrac a2

and

h

, which gives

h = \tfrac{\sqrt 3}{2}a

. A regular hexagon is six of those triangles meeting at its centre, each with the hexagon's side.

Definition

Equilateral triangle of side $a$ : height $\dfrac{\sqrt 3}{2}a$ , area $\dfrac{\sqrt 3}{4}a^2$ , perimeter $3a$ .
Regular hexagon of side $a$ : six equilateral triangles, so area $\dfrac{3\sqrt 3}{2}a^2$ and perimeter $6a$ .
Going back: from the area, $a^2 = \dfrac{4A}{\sqrt 3}$ ; from the height, $a = \dfrac{2h}{\sqrt 3}$ .

Equilateral triangle and hexagon

A_{\triangle} = \dfrac{\sqrt 3}{4}a^2,\qquad A_{\text{hex}} = 6 \times \dfrac{\sqrt 3}{4}a^2 = \dfrac{3\sqrt 3}{2}a^2

Worked example

Find the height and area of an equilateral triangle of side 10 cm, and the area of a regular hexagon of side 10 cm.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Area and Perimeter · Quadrilaterals and Regular Polygons

A regular hexagon has side

4

cm. Find its area: (use

\sqrt{3} = 1.732

)

Height has a 2 under it, area a 4

For side 10 the height is

5\sqrt 3

and the area

25\sqrt 3

. Swapping the two denominators gives an area of

50\sqrt 3

, twice too big.

Six triangles, not three

A regular hexagon of side 10 holds six equilateral triangles of side 10, so its area is

150\sqrt 3

. Counting three gives half the answer.

Drill 3 more on equilateral triangles and the regular hexagon

Concept 4 of 4: Angles and perimeter of a regular polygon

Lines from one corner of an

n

-sided polygon cut it into

n - 2

triangles, each with 180° of angle. In a regular polygon all sides and all angles are equal, so the total is shared out equally, and walking round it turns you through 360° in all.

Definition

Sum of interior angles $= (n - 2) \times 180°$ .
Each interior angle of a regular polygon $= \dfrac{(n - 2) \times 180°}{n}$ .
Each exterior angle $= \dfrac{360°}{n}$ , and interior + exterior $= 180°$ . From an exterior angle, $n = \dfrac{360°}{\text{exterior}}$ .
Perimeter of a regular $n$ -gon of side $a$ is $na$ . Equal perimeters: $n_1 a_1 = n_2 a_2$ .

Polygon angles

\text{sum} = (n - 2) \times 180°,\qquad \text{exterior} = \dfrac{360°}{n}

Worked example

For a regular octagon, find the sum of the interior angles, each interior angle and each exterior angle.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Area and Perimeter · Quadrilaterals and Regular Polygons

Interior angle of a regular pentagon is:

Sum or each

(n - 2) \times 180°

is the total. For an octagon that is 1080°; each angle is 135°. Options often carry both.

Exterior angles always total 360°

Whatever the number of sides, the exterior angles of a polygon add to 360°. Going from an angle to the number of sides is quickest through the exterior angle.

Drill 5 more on angles and perimeter of a regular polygon

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Rhombus and kite: working from the diagonals
Rhombus
$A = \tfrac12 d_1 d_2,\qquad \text{side} = \sqrt{\left(\tfrac{d_1}{2}\right)^2 + \left(\tfrac{d_2}{2}\right)^2}$
Trapezium, parallelogram and any quadrilateral
Trapezium and quadrilateral
$A_{\text{trap}} = \tfrac12(a + b)h,\qquad A_{\text{quad}} = \tfrac12 d(h_1 + h_2)$
Equilateral triangles and the regular hexagon
Equilateral triangle and hexagon
$A_{\triangle} = \dfrac{\sqrt 3}{4}a^2,\qquad A_{\text{hex}} = 6 \times \dfrac{\sqrt 3}{4}a^2 = \dfrac{3\sqrt 3}{2}a^2$
Angles and perimeter of a regular polygon
Polygon angles
$\text{sum} = (n - 2) \times 180°,\qquad \text{exterior} = \dfrac{360°}{n}$

Watch out for (8)

Halve the diagonals first
→ Rhombus and kite: working from the diagonals
Half the product
→ Rhombus and kite: working from the diagonals
Each height goes with its own sides
→ Trapezium, parallelogram and any quadrilateral
The slant side is not the height
→ Trapezium, parallelogram and any quadrilateral
Height has a 2 under it, area a 4
→ Equilateral triangles and the regular hexagon
Six triangles, not three
→ Equilateral triangles and the regular hexagon
Sum or each
→ Angles and perimeter of a regular polygon
Exterior angles always total 360°
→ Angles and perimeter of a regular polygon

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