Area Postulate & Decomposition
This postulate guarantees that a region’s area equals the sum of its non-overlapping parts.
What is the Area Postulate?
A square has area 81. What is its side length?
9
Each grid square has side 5 units. What is the area of one square?
25 square units
A triangle has sides 4, 3, and 5. Classify it.
Right triangle.
Convert 4 square feet to square inches.
4 × 144 = 576 in²
A figure is divided into regions of area 12, 18, 20, and 30. What is the total area?
80 square units
Squares on the legs of a right triangle have areas 49 and 144. What is the hypotenuse length?
√(49 + 144) = √193 ≈ 13.89300
A region covers 32 squares of size 10m × 10m. What is the total area?
32 × 100 = 3,200 m²
A triangle’s sides are doubled. What happens to its area?
What is it becomes four times as large?
A square has side 36 inches. What is its perimeter in feet?
144 inches = 12 feet
A “triangle” has base 20 and height 12 giving area 120, but summing parts gives 118. What likely caused this difference?
The figure is not truly a triangle (sides not collinear or slight gaps or overlaps).
A triangle has sides 10, 24, 26. Is it right?
Yes. 10² + 24² = 100 + 576 = 676 = 26².
T/F If a region has greater perimeter, it must have greater area.
False. Perimeter does not determine area.
A parallelogram has base 15 and height 8. Find its area.
120
A trapezoid has bases 8 and 14 and height 6. Find its area.
½(8+14)(6) = 66
Two triangles have equal areas. One has twice the base of the other. What must be true about their altitudes?
What is the altitude must be half as large?
The diagonal of a square is 10. Find its area.
Side = 10/√2 = 5√2
Area = (5√2)² = 50
A 12×12 square contains two smaller squares of areas 36 and 25. How can you compare the remaining regions?
Subtract known areas from total (144 − 36 − 25 = 83) and analyze remaining pieces.
Why are the legs of a right triangle altitudes?
They are perpendicular, satisfying the definition of altitude.
A triangle has sides 5, 5, and 10. Does it have area?
What is no, it is not a triangle
A quadrilateral appears to be a parallelogram but opposite sides are not parallel. How does this affect area?
Base × height no longer applies; area calculation would be incorrect.
Three squares have areas 200, 72, and 128. The sides of the 3 squares form a triangle. Is the triangle right?
Yes. 72 + 128 = 200.
Name two strategies to find the area of a cross-shaped grid figure.
Decompose into rectangles OR subtract corner triangles from full square.
Which has greater area: right triangle (leg 6, hyp 10), square (side 4), or trapezoid (bases 2,3 height 10)?
Right triangle = 24; square = 16; trapezoid = 25 → trapezoid largest.
Describe a situation where a figure has larger perimeter but smaller area than another.
A long thin rectangle vs. a compact square.