Perimeter and Area of Polygons - SSAT Upper Level Quantitative

A regular pentagon has a perimeter of 45 cm. If the apothem (distance from center to the middle of any side) is 6.2 cm, what is the area?
Answer: 139.5 cm² When you encounter a regular polygon with given perimeter and apothem, you're looking at an area calculation that uses the formula: Area = $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$ . This formula works because any regular polygon can be divided into congruent triangles from the center, where each triangle has a base equal to one side length and height equal to the apothem. Since Area = $\frac{1}{2} \times \text{base} \times \text{height}$ for triangles, and you have multiple triangles, this simplifies to the perimeter-apothem formula. With a perimeter of 45 cm and apothem of 6.2 cm: Area = $\frac{1}{2} \times 45 \times 6.2 = \frac{1}{2} \times 279 = 139.5$ cm². This confirms answer D. The wrong answers represent common calculation errors: A) 93 cm² likely comes from miscalculating $\frac{45 \times 6.2}{3}$ instead of dividing by 2, possibly confusing triangle area formulas. B) 279 cm² is the result of forgetting to multiply by $\frac{1}{2}$ entirely—this is just perimeter times apothem without the area formula's division. C) 186 cm² might result from computational errors in the multiplication or using incorrect values. Study tip: Memorize that for any regular polygon, Area = $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$ . This formula appears frequently on standardized tests and eliminates the need to calculate individual side lengths or use more complex polygon formulas. Always double-check that you've included the $\frac{1}{2}$ factor.
A trapezoid has parallel sides of lengths 10 cm and 22 cm. If the height is 8 cm and the non-parallel sides each have length 10 cm, what is the perimeter?
Answer: 52 cm The perimeter is the sum of all four sides: $10 + 22 + 10 + 10 = 52$ cm. The height is not a side, so it is not part of the perimeter. The givens are consistent: each leg covers a horizontal overhang of $(22 - 10) ÷ 2 = 6$ cm, and $6^2 + 8^2 = 36 + 64 = 100 = 10^2$ . 40 cm adds the two parallel sides and the height ( $10 + 22 + 8$ ). 42 cm leaves out one of the non-parallel sides ( $10 + 22 + 10$ ). 50 cm uses the height in place of one of the non-parallel sides ( $10 + 22 + 8 + 10$ ). Strategy tip: Perimeter means the distance around the outside. List every side by name before adding, and leave out measurements such as heights that are not sides.
A regular hexagon is inscribed in a circle with radius 8 cm. What is the perimeter of the hexagon?
Answer: 48 cm In a regular hexagon inscribed in a circle, each side length equals the radius of the circle. This is because connecting the center to adjacent vertices forms equilateral triangles. Since the radius is 8 cm, each side is 8 cm. The perimeter is 6 × 8 = 48 cm. Choice A incorrectly uses 4 cm as the side length. Choice C uses the apothem instead of the side length. Choice D applies an incorrect formula involving $\sqrt{3}$ .
A rectangular garden is expanded by increasing its length by 4 meters and its width by 2 meters. If the original rectangle had length 12 meters and width 8 meters, by how much does the area increase?
Answer: 64 square meters Original area = 12 × 8 = 96 m². New dimensions are 16 m × 10 m, so the new area = 16 × 10 = 160 m². The increase is 160 − 96 = 64 m². Adding only the two side strips (4 × 8 + 2 × 12 = 56) misses the 4 × 2 corner piece where both strips overlap, so 56 is 8 short. Using only the strip added along the length (4 × 8 = 32) ignores the width increase, and multiplying the total added length and width by the original width ((4 + 2) × 8 = 48) does not describe either strip.
A rhombus has diagonals of length 16 cm and 12 cm. What is the area of the rhombus?
Answer: 96 cm² The area of a rhombus equals half the product of its diagonals: Area = ½ × 16 × 12 = 96 cm². Choice A incorrectly multiplies the full diagonals (16 × 12). Choice C uses the wrong formula, dividing by 4 instead of 2. Choice D incorrectly adds the diagonals then multiplies by 3 (16 + 12) × 3.
A parallelogram has adjacent sides of 15 cm and 20 cm, with an included angle of 60°. What is the area of the parallelogram?
Answer: 150 $\sqrt{3}$ cm² When you encounter a parallelogram problem with two sides and an included angle, you're dealing with area calculation using the sine formula. The area of a parallelogram equals the product of two adjacent sides times the sine of the included angle: $A = ab\sin(\theta)$ . Let's apply this formula with the given values: adjacent sides of 15 cm and 20 cm, with a 60° angle between them. So $A = 15 \times 20 \times \sin(60°)$ . Since $\sin(60°) = \frac{\sqrt{3}}{2}$ , we get $A = 300 \times \frac{\sqrt{3}}{2} = 150\sqrt{3}$ cm². Looking at the wrong answers: Choice A (300 cm²) represents the trap of simply multiplying the two sides without accounting for the angle—this would only work if the angle were 90°. Choice B (260 cm²) might result from incorrectly using $\sin(60°) = 0.866$ and rounding $300 \times 0.866 = 259.8$ to 260, but this loses the exact answer. Choice D ( $300\sqrt{3}$ cm²) makes the error of multiplying by $\sqrt{3}$ instead of $\frac{\sqrt{3}}{2}$ , essentially doubling the correct result. The key strategy here is recognizing that parallelogram area problems with an angle always require the sine formula, not just base times height. Also, remember that $\sin(60°) = \frac{\sqrt{3}}{2}$ exactly—this is a crucial trigonometric value to memorize for the SSAT.
An isosceles triangle has two equal sides of length 13 cm and a base of length 10 cm. What is the area of the triangle?
Answer: 60 cm² To find the area, we need the height. The height bisects the base, creating two right triangles with hypotenuse 13 cm and base 5 cm. Using the Pythagorean theorem: height = $\sqrt{13² - 5²} = \sqrt{169 - 25} = \sqrt{144} = 12$ cm. Area = ½ × 10 × 12 = 60 cm². Choice B incorrectly uses 13 as the height. Choice C uses the full base length instead of half. Choice D omits the factor of ½.

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