10. Geometry

Perimeter and Area of Polygons

10. Geometry

Perimeter and Area of Polygons: 동영상 및 연습문제

주제 요약

Perimeter is the distance around the outside of a polygon, found by adding all side lengths. For rectangles, a useful shortcut is $P=2L+2W$ . For irregular or compound shape figures, first find any missing side lengths by comparing parallel sides and using addition or subtraction, then trace around the figure so each side is counted exactly once. Perimeter uses the same linear units as the side lengths.

Area is the amount of space inside a polygon, and it is always measured in square units. Key formulas include rectangle $A=L\times W$ , square $A=S^2$ , triangle $A=\frac{1}{2}bh$ , parallelogram $A=bh$ , and trapezoid $A=\frac{1}{2}(B_1+B_2)h$ . In Perimeter and Area of Polygons, compound figures are handled by splitting them into known shapes and adding the smaller areas.

개념

Perimeter

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문제

Find the perimeter of the isosceles triangle (two sides have equal length).
$Isosceles triangle with two equal sides of 16 inches and a base of 21 inches labeled for perimeter calculation.$

$106$ in

$74$ in

$37$ in

$53$ in

문제

Find the perimeter of a rectangle with length 12 yd and width 8 yd.

$20$ yd

$40$ yd

$24$ yd

$32$ yd

예시

Perimeter Example 1

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예시

Perimeter Example 2

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개념

Area

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문제

Find the area of the figure.
$Square with each side measuring 12 centimeters, labeled for calculating area.$

$144$ cm²

$48$ cm²

$576$ cm²

$24$ cm²

문제

Find the area of the figure.
$Equilateral triangle with side length 3 feet and height 3.5 feet, labeled for area calculation.$

$10$ ft²

$10.5$ ft²

$5.25$ ft²

$5$ ft²

문제

Find the area of the figure.
$Trapezoid with parallel sides 6 mi and 9 mi, height 5.5 mi, labeled for area calculation.$

$41.25$ mi²

$82.5$ mi²

$49.5$ mi²

$33$ mi²

예시

Area Example 3

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예시

Area Example 4

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학생들이 이 주제에 대해 묻는 질문은 다음과 같습니다:

To find the perimeter of a rectangle, you add the lengths of all four sides. Since opposite sides of a rectangle are equal, you can use the formula $P = 2 L + 2 W$ , where $L$ is the length and $W$ is the width. For example, if the length is 8 inches and the width is 5 inches, the perimeter is $2 \times 8 + 2 \times 5 = 26$ inches. Always ensure all side lengths are in the same units before calculating, and include units in your final answer.

The area of a triangle is found using the formula $A = \frac{1}{2} b h$ , where $b$ is the base length and $h$ is the height (the perpendicular distance from the base to the opposite vertex). For example, if the base is 2 cm and the height is 6 cm, the area is $\frac{1}{2} 26 = 6$ cm $c^{2}$ . Remember, the height must be perpendicular to the base, and it may not always be one of the triangle's sides.

To calculate the area of a compound polygon, first divide the shape into simpler known shapes such as rectangles, triangles, or trapezoids. Then, find the area of each smaller shape using the appropriate formulas. Finally, add all these areas together to get the total area of the compound polygon. For example, if a compound shape can be split into a rectangle and a triangle, calculate each area separately and sum them. This method helps manage irregular shapes by breaking them down into manageable parts. Always ensure units are consistent and express the final area in square units.

Perimeter and area measure different properties of polygons. The perimeter is the total distance around the outside of the polygon, found by adding all side lengths. It is measured in linear units like inches or meters. Area, on the other hand, measures the amount of space inside the polygon and is expressed in square units, such as square inches or square meters. While perimeter involves addition of side lengths, area involves multiplication of dimensions, often using specific formulas depending on the polygon type. Understanding both concepts is essential for solving geometry problems involving polygons.

When calculating the perimeter of irregular polygons, some side lengths may be missing. To find these, use the properties of parallel sides and the total lengths given. For example, if two vertical sides together equal a known height, add their lengths to find the missing side. Similarly, if the total horizontal length is known, subtract the known part from the total to find the unknown side. This approach uses addition or subtraction of parallel side lengths. After finding all missing sides, add all side lengths once to get the perimeter. Tracing the polygon while adding helps ensure no side is counted twice or missed.