Unit 9: Area, Perimeter, and Geometric Probability
Turning a shape's dimensions into the number of square units it covers, then using that same area to measure the odds of landing on a target.
7 topics · 24 questions
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Area and Perimeter of Triangles and Parallelograms
- How it worksHow does slicing a triangle off one end of a parallelogram and reattaching it to the other prove the area formula?
- Worked exampleFind the area of a triangle with base 14 and height 9.
- Worked exampleFind the perimeter of a parallelogram with sides 6 and 11.
- Common mistakeWhy doesn't a parallelogram's slanted side count as its height in the area formula?
Your turn1 of 2
Find the area of a triangle with base 22 and height 21.Area of Trapezoids, Kites, and Rhombuses
- How it worksHow does a rhombus's area come from half the product of its two diagonals?
- Worked exampleFind the area of a trapezoid with parallel sides 7 and 13 and height 5.
- Worked exampleFind the area of a kite with diagonals of 9 and 16.
- Common mistakeWhy doesn't a rhombus's area formula work from its side length alone, without an angle or a diagonal?
Your turn1 of 2
Find the area of a trapezoid with parallel sides 15 and 14 and height 19.Area of Regular Polygons
- How it worksHow does the apothem, the distance from center to a side, become the height of each triangle slice?
- Worked exampleFind the area of a regular hexagon with side length 6.
- Common mistakeWhy isn't the apothem the same length as the polygon's radius to a vertex?
Your turn1 of 2
Find the area of a regular hexagon with side length 21.Area of Composite Figures
- How it worksHow do you add and subtract the areas of simple shapes to find the area of an irregular one?
- Worked exampleFind the area of an L-shaped figure made from a 10 by 8 rectangle with a 4 by 3 rectangle removed from one corner.
- Going furtherWhat happens to a composite figure's area calculation when one of its pieces is a semicircle instead of a rectangle?
Your turn1 of 2
An L-shaped figure is made from a 17 by 16 rectangle with a 8 by 4 rectangle removed from one corner. Find the area of the L-shaped figure.Geometric Probability with Length and Area
- How it worksHow does dividing a target region's area by the total region's area turn geometry into a probability?
- Worked exampleFind the probability that a randomly thrown dart lands in a 4 by 4 square drawn inside a 10 by 10 square target.
- Worked exampleFind the probability that a point picked randomly on a 12 centimeter segment falls within 3 centimeters of its midpoint.
- Common mistakeWhy doesn't geometric probability work the same way when outcomes aren't spread evenly across the region?
Your turn1 of 2
A dart lands randomly somewhere in a 16 by 16 square target. Find the probability that it lands within a smaller 5 by 5 square drawn inside it.Modeling Real Objects with Geometric Shapes and Density
- How it worksHow does dividing a population by a region's area turn a messy real map into a single density number?
- Worked exampleFind the population density of a rectangular park that measures 400 by 250 meters and holds 2000 people.
- Going furtherWhat happens to a city's calculated density when its border is modeled as a circle instead of a rectangle?
Your turn1 of 2
A rectangular park measures 379 by 360 meters and holds 4695 people. Find its population density, in people per square meter.Maximizing Area for a Fixed Perimeter
- How it worksHow does shifting a rectangle's shape toward a square while holding its perimeter fixed increase its area?
- Worked exampleFind the maximum possible area of a rectangle with a perimeter of 40 meters.
- Common mistakeWhy doesn't a longer, thinner rectangle ever beat a square for area at the same perimeter?
Your turn1 of 2
Find the maximum possible area of a rectangle with a perimeter of 72 meters.