1.2.12 · D5Basic Geometry
Question bank — Area — triangle, parallelogram, trapezium, composite shapes
Before starting, one word we lean on everywhere: the perpendicular height is the straight-line distance from a base to the opposite side, measured at a right angle to that base — never a slant edge. Almost every trap below is a disguised test of whether you kept perpendicular in mind.
True or false — justify
Two triangles with the same base and the same perpendicular height always have the same area.
True — area is , and it depends only on those two numbers, so the apex can slide sideways as far as you like without changing anything.
A parallelogram with base 10 and slant side 7 has area 70.
False — area is base times perpendicular height, and the slant side of 7 is longer than the true height, so the real area is less than 70.
If you double the base of a triangle but halve its height, the area is unchanged.
True — area is , and , so the two changes cancel exactly.
Every trapezium formula works even when the two parallel sides are equal.
True — if the shape is a parallelogram and , which is the correct parallelogram area, so the formula degrades gracefully.
The height of a triangle must lie inside the triangle.
False — for an obtuse triangle the foot of the perpendicular lands outside the base, so the height sits outside the shape yet still measures the same .
A rectangle is a special case of both a parallelogram and a trapezium.
True — its opposite sides are parallel and equal (parallelogram), and it also has a pair of parallel sides with the perpendicular height equal to the other side (trapezium).
Cutting a shape and rearranging the pieces can change its area.
False — rearranging pieces without overlap or gap preserves total area; this invariance is exactly why we can turn a parallelogram into a rectangle.
If two shapes have equal area, they must have equal perimeter.
False — a long thin rectangle and a near-square can share the same area but have very different perimeters, since area and perimeter respond differently to stretching (see 1.2.11-Perimeter).
The average of a trapezium's two parallel sides equals the width of a rectangle with the same height and area.
True — , so is literally the width of that equal-area rectangle.
Spot the error
"The triangle has sides 6, 8, 10, so its area is ."
Wrong — 10 is the hypotenuse, not a height perpendicular to 8; the two legs 6 and 8 are perpendicular, so area is .
"This parallelogram leans a lot, so I'll use the longer slant side as height to get a bigger, safer area."
Wrong — the more it leans, the smaller the perpendicular height and the smaller the area; using the slant overestimates rather than being "safe."
"For the trapezium I added the two parallel sides, multiplied by height, and got the area."
Wrong — you forgot the factor ; the trapezium is half the parallelogram built from two copies, so it is .
"To find an L-shape's area I measured the outer bounding rectangle: ."
Wrong — the bounding rectangle includes the missing corner, so you must subtract the cut-out piece rather than reporting the whole box.
"The frame's area is the outer rectangle minus the inner one, but I subtracted their perimeters instead."
Wrong — area and perimeter are different quantities; subtract areas (), never lengths, to find the material of the frame.
"I used 4 cm as the height because it's one of the slant sides."
Wrong — a slant side only counts as the height when it is perpendicular to the base; otherwise you must find the true perpendicular distance, often via 1.3.5-Pythagorean-theorem or 2.1.4-Trigonometric-ratios.
"Two overlapping regions cover a big area, so I added both areas to get the total."
Wrong — the overlap gets counted twice; the true covered area is minus the overlapping region once.
Why questions
Why is the triangle formula exactly half of base × height?
Because two identical triangles fit together into a parallelogram of area , so one triangle is half of it.
Why does the slant side of a parallelogram not appear in its area formula?
Because sliding the corner triangle across turns the shape into a rectangle of base and height ; the slant only sets how far it leans, not how much floor it covers.
Why do we need the sine of the angle when only the slant side and base are given?
Because extracts the perpendicular part of the slant side, giving , which is the vertical squeeze that matters for area — this is the trig ratio "opposite over hypotenuse" from 2.1.4-Trigonometric-ratios.
Why does the trapezium formula use the average of the parallel sides?
Because is the width of an equal-area rectangle: the shape widens from to linearly, so its effective width is the midpoint value.
Why can any side of a triangle be chosen as the base?
Because area is a single number; each base pairs with its own perpendicular height so that always returns the same value.
Why does integration get called "area under a curve," linking back to this chapter?
Because a curve region is sliced into thin near-rectangles whose areas are summed — the same "tile it and add" idea, taken to infinitely thin strips (see 3.4.2-Integrationas-area).
Why must composite-shape dimensions sometimes be deduced rather than read off directly?
Because an L-shape's hidden edge length equals the difference of the outer edges, so opposite sides of the full outline must add up consistently.
Edge cases
What is the area of a triangle whose three points lie on one straight line?
Zero — the height collapses to , so ; a degenerate triangle covers no space.
What happens to a parallelogram's area as its lean angle approaches ?
It approaches , because ; the shape flattens into a line with no area.
What does the trapezium formula give when one parallel side shrinks to zero length?
It becomes , the triangle formula — a triangle is a trapezium with a zero-length "top."
Can a composite shape's cut-out be larger than the outer shape?
No — a physical cut-out must fit inside the outer region, so subtracting a larger "hole" would give a negative area, signalling a measurement error.
What is the area of a rectangle with zero width?
Zero — one dimension is , so ; it has degenerated into a line segment with length but no area.
If a parallelogram's angle reaches , what shape and area result?
It becomes a rectangle, and , so the area is simply base × slant side — the maximum area for those two side lengths.
Recall Quick self-check
The one idea behind almost every trap here ::: Distinguish a perpendicular height from any slant length; area always uses the perpendicular. Why degenerate cases (zero height, collinear points) are worth checking ::: They confirm a formula behaves sensibly at its limits and reveal when a shape has silently collapsed.