Justify polygon angle sums and area formulas by partitioning into simpler shapes.
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Try
A trapezoid with parallel bases 7 and 13 and perpendicular height 4 has area 40. The sloping side is longer than the height; replacing the height by that side would overestimate the area.
Predict
Predict why a polygon with more sides has a larger interior-angle sum. In a convex polygon, diagonals from one vertex remain inside it and partition it into triangles. The number of pieces is n−2, not n. Convexity means every segment between two points of the polygon stays in it.
Experiment
Draw the four triangles in a convex hexagon by fanning diagonals from one vertex. Count the triangles and check which triangle angles combine at the fan vertex. Separately redraw the trapezoid as two congruent pieces that combine into a parallelogram; do not overlap the interiors.
A trapezoid with parallel bases 13 and 7 and perpendicular height 4, all drawn to the same scale. Its area is 40 square units.
Understand
Worked argument: two congruent trapezoids, one rotated half a turn, join to form a parallelogram of base a+b and height h. The parallelogram area is (a+b)h, so each trapezoid has area (a+b)h/2. For a=7,b=13,h=4 this gives 20×4/2=40. The same formula also works when the trapezoid is a rectangle.
Invalid decomposition: summing areas of triangles that overlap counts the overlap twice. A valid partition covers the region exactly, with intersections only on boundaries. If subtracting a hole instead, specify that the hole lies within the outer region and use consistent square units.
Apply
Distinguish an angle sum from the size of one angle: dividing by n requires an equiangular polygon, not just a convex one. For area, draw the perpendicular height even when its foot lies outside the chosen side segment. State whether an answer is a length, area or angle.
Continue
Depth: simple nonconvex polygons also have interior-angle sum (n−2)×180°, but a fan from an arbitrary vertex may leave the polygon. A more careful triangulation proves that extension. Self-intersecting paths need separate conventions and are not covered by the usual simple-polygon formula.
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Practice and reasoning
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Find the interior-angle sum of a convex hexagon in degrees.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A trapezoid has parallel bases 7 and 13 cm and height 4 cm. Find its area in cm².
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For area=bh of a parallelogram, h is:
For a convex polygon traversed once, the sum of the magnitudes of its exterior turns (one at each vertex) is:
Prove the interior-angle sum of a convex n-gon is (n−2)×180°.
Structure your argument and preserve revisions
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Derive the trapezoid area formula for bases a,b and perpendicular height h by making two copies.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
