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Interior and exterior angle sums, regular polygons, and the areas of trapezoids, rhombi and composite figures.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you cut a polygon into triangles from one corner and read off the sum of its interior angles, and you see why the exterior angles of every polygon add to $360^\circ$ however many sides it has. You use both facts on regular polygons, forwards to find an angle and backwards to count the sides. Then you find areas: a trapezoid from its two parallel sides and its height, a rhombus from its diagonals, and a composite figure by splitting it into pieces whose areas you already know.
You know that the three angles of a triangle add to $180^\circ$, that angles on a straight line add to $180^\circ$ and that angles round a point add to $360^\circ$. You can find the area of a rectangle and of a triangle. Every result in this lesson is one of those four facts applied to a shape with more sides than a triangle.
Polygon: a closed figure made of straight sides. An $n$-gon has $n$ of them.
Regular polygon: every side the same length and every angle the same size. Both conditions matter — a rhombus has equal sides but is not regular.
Interior angle: an angle inside the polygon, at one corner. Exterior angle: the angle between one side and the extension of the side next to it — the amount you turn as you walk round the corner. The two together make a straight line.
Diagonal: a segment joining two corners that are not next to each other.
Trapezoid: a quadrilateral with one pair of parallel sides, called its bases; the height is the perpendicular distance between them, not the length of a slanted side.
Rhombus: a quadrilateral with four equal sides. Its diagonals cross at right angles.
Composite figure: a shape built from simpler ones stuck together, and measured by measuring the pieces.
Draw every diagonal from one corner of an $n$-sided polygon and it falls into $n - 2$ triangles, so the interior angles add to $(n - 2) \times 180^\circ$: a hexagon gives $4 \times 180 = 720^\circ$. Walk once round any polygon and you turn through a full circle, so the exterior angles add to $360^\circ$ no matter how many sides there are. In a regular polygon those turns are equal, so a regular octagon turns $360 \div 8 = 45^\circ$ at each corner and its interior angles are $180 - 45 = 135^\circ$ each. Run that backwards to count sides: an interior angle of $150^\circ$ leaves an exterior angle of $30^\circ$, and $360 \div 30 = 12$ sides. For area, a trapezoid is the average of its two parallel sides times the height, so bases $7$ and $13$ with height $6$ give $10 \times 6 = 60$. A rhombus is half the product of its diagonals: $10$ and $24$ give $120$. A composite figure is measured piece by piece: a $10$ by $6$ rectangle with a right triangle of legs $6$ and $4$ on one end is $60 + 12 = 72$.
Another way: picture
A hexagon with three diagonals drawn from one corner, splitting it into four triangles. Each triangle is labelled $180^\circ$, and $4 \times 180 = 720^\circ$ is written underneath.
Another way: story
Walk the outside of a field, whatever its shape, and end up facing the way you started. Every corner turned you a little, and altogether you turned exactly one full circle. That is the whole proof that the exterior angles add to $360^\circ$, and it never mentions the number of sides — which is why the answer does not depend on it.
Multiplying by the number of sides instead of two fewer. A hexagon splits into four triangles, not six, so its angles total $720^\circ$ and not $1080^\circ$. Draw the diagonals from one corner and count: the two sides that meet at that corner are not the base of any triangle.
Dividing $360$ by the interior angle. The exterior angles are what add to $360^\circ$, so a regular polygon with interior angle $150^\circ$ has $360 \div 30 = 12$ sides, not $360 \div 150$. Convert to the exterior angle first, every time.
Losing the halving. A trapezoid uses the average of its bases and a rhombus uses half the product of its diagonals. Forgetting either one doubles the answer — which is why an area exactly twice the right size is such a common wrong answer that it has its own feedback line.
The interior and exterior angle at a corner make a straight line, so the exterior angle is $180 - 150 = 30^\circ$.
Switch to exterior angles: they are the ones with a fixed total.
The exterior angles add to $360^\circ$ and are all equal, so there are $360 \div 30$ of them.
$= 12$ sides.
Check: $(12 - 2) \times 180 \div 12 = 150^\circ$.
The diagonals cross at right angles and cut each other in half, so the rhombus sits inside a $10$ by $24$ rectangle, touching the middle of each side.
It fills half of that rectangle: $(10 \times 24) \div 2 = 120$.
Half the product of the diagonals.
Split it into the two pieces you already know.
Rectangle: $10 \times 6 = 60$.
Triangle: $6 \times 4 \div 2 = 12$, so the figure is $60 + 12 = 72$.
What do the interior angles of a polygon with $11$ sides add up to, in degrees?
answer
A regular polygon has $18$ equal sides. How many degrees is each of its exterior angles?
answer
Each interior angle of a regular polygon measures $165^\circ$. How many sides has the polygon?
Answer:
A trapezoid has parallel sides of $16$ and $23$, and the distance between them is $12$. What is its area?
Answer:
A tiler opens a crate of identical tiles. Each one is a regular polygon with $6$ equal sides. Laid flat, with no gaps and no overlaps and no tile cut, can tiles of this one shape cover a whole floor?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
The diagonals of a rhombus are $24$ and $3$ long. What is its area?
Answer:
You can find polygon angle sums, work back from a regular polygon's angles to its number of sides, and find the areas of trapezoids, rhombi and composite figures. Explain why the exterior angles of a polygon always add to $360^\circ$, whatever the number of sides, and say what has to be halved when you are given the diagonals of a rhombus.
8. Your turn: a $10$ by $6$ rectangle with a right triangle of legs $6$ and $4$ joined to one end, step 3