E5.5 - Irregular polygons (Extended)

E5.5 - Irregular polygons (Extended)

A house-shaped pentagon can have two right angles at its base and another at the roof peak. What must the two remaining angles add to? E5.5 uses angle sums and local angle facts together, including cases where the polygon's angles are unequal.

Irregular angle sums

An irregular polygon does not have both equal sides and equal angles. Its angles need not be equal, but the total still comes from splitting the shape into triangles.

Angle sum of an irregular polygon

A simple nn-sided polygon has interior angle sum (n−2)×180∘(n-2)\times180^\circ. “Simple” means its sides do not cross. The result includes concave polygons, which have an inward corner with an interior angle greater than 180∘180^\circ. Not on the supplied formula list: learn the triangle argument.

For a concave shape, choose diagonals that stay inside the polygon; they still divide it into n−2n-2 triangles. Count the angle inside the shape at an inward corner, including its reflex part.

Worked example 1: an unknown pentagon angle

Question. The interior angles of a pentagon, in order, are 92∘,118∘,105∘,130∘92^\circ,118^\circ,105^\circ,130^\circ and x∘x^\circ. Calculate xx.

Plan. Subtract the four known angles from the pentagon total.

S=3×180∘=540∘five sidesx=540−(92+118+105+130)remaining interior angle=95\begin{aligned} S&=3\times180^\circ=540^\circ &&\text{five sides}\\ x&=540-(92+118+105+130) &&\text{remaining interior angle}\\ &=95 \end{aligned}

Answer. x=95x=95, so the angle is 95∘95^\circ.

Check. 92+118=21092+118=210 and 105+130+95=330105+130+95=330; together they make 540540.

The next pentagon has an inward corner. Its reflex angle still belongs in the total.

Worked example 2: an inward corner

Question. A concave pentagon has interior angles 70∘,80∘,85∘,75∘70^\circ,80^\circ,85^\circ,75^\circ and r∘r^\circ. Calculate its reflex angle r∘r^\circ.

Plan. Use the same total, keeping the interior angle at the inward corner.

r=540−(70+80+85+75)pentagon angle sum=540−310=230\begin{aligned} r&=540-(70+80+85+75) &&\text{pentagon angle sum}\\ &=540-310=230 \end{aligned}

Answer. r=230r=230, so the reflex angle is 230∘230^\circ.

Check. The smaller angle outside that corner is 360∘−230∘=130∘360^\circ-230^\circ=130^\circ; the interior is correctly larger than 180∘180^\circ.

The angle total does not tell you that all the angles are equal.

Watch out

Dividing 540∘540^\circ by 5 gives the interior angle only if the pentagon is equiangular. For an irregular polygon, add the actual given angles.

Keep the reflex angle in the total in this new case.

Exterior angles

For a convex polygon, one exterior turn at every vertex adds to 360∘360^\circ, even when the turns are unequal. Each is supplementary to the interior angle at that same vertex.

A complete set of exterior turns

Walk around a convex polygon in one direction and take one exterior turning angle at each vertex. These turns total 360∘360^\circ. For an irregular polygon, they need not have the same size.

Choose either the complete-turn total or the local straight-line relationship, depending on which information you have. The positive-turn calculation here is for convex shapes; at an inward corner, the direction of turning reverses.

Worked example 3: one missing turn

Question. Four exterior turns of a convex pentagon are 50∘,80∘,65∘50^\circ,80^\circ,65^\circ and 75∘75^\circ. Calculate the fifth exterior turn and its adjacent interior angle.

Plan. Complete 360∘360^\circ, then use the straight angle at that vertex.

e=360−(50+80+65+75)=90complete turni=180−90=90straight angle\begin{aligned} e&=360-(50+80+65+75)=90 &&\text{complete turn}\\ i&=180-90=90 &&\text{straight angle} \end{aligned}

Answer. Exterior 90∘90^\circ, interior 90∘90^\circ.

Check. The other interior angles are 130∘,100∘,115∘,105∘130^\circ,100^\circ,115^\circ,105^\circ. With 90∘90^\circ, they total 540∘540^\circ.

The next example mixes interior and exterior information, so first put all the data into the same form.

Worked example 4: mixed angle information

Question. A convex pentagon has interior angles 105∘,125∘,100∘105^\circ,125^\circ,100^\circ at three vertices. Its exterior turn at a fourth vertex is 70∘70^\circ. Calculate the exterior turn at the fifth vertex.

Plan. Convert the three interior angles to exterior turns, then complete one revolution.

e1=180−105=75straight anglee2=180−125=55straight anglee3=180−100=80straight anglee5=360−(75+55+80+70)=80complete turn\begin{aligned} e_1&=180-105=75 &&\text{straight angle}\\ e_2&=180-125=55 &&\text{straight angle}\\ e_3&=180-100=80 &&\text{straight angle}\\ e_5&=360-(75+55+80+70)=80 &&\text{complete turn} \end{aligned}

Answer. 80∘80^\circ.

Check. The last two interior angles are 110∘110^\circ and 100∘100^\circ. All five interior angles total 105+125+100+110+100=540105+125+100+110+100=540 degrees.

“Exterior” refers to the turn formed by extending a side, not every angle drawn outside a shape.

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