Perimeter & Area

MATH 102Contemporary Math

S. Rook

Overview

• Section 10.3 in the textbook:– Perimeter & area– Pythagorean theorem

Perimeter & Area

Perimeter & Area in General

• Perimeter: the sum of the lengths of the sides of a polygon– i.e. the length around (“rim”) the polygon– Measured using the same units as the sides

• Area: the amount of space inside of the polygon– Measured in square units of the sides

• On the next few slides are formulas for perimeter and area for common polygons

Parallelogram

• Recall the shape and characteristics of a parallelogram

• No special formula for perimeter– Just add the lengths of all the sides

• For a parallelogram with height h and base b, area = h x b– The height is a straight vertical line– See page 471 for theory

Trapezoid

• Recall the shape and characteristics of a trapezoid

• No special formula for perimeter– Just add up the sides

• For a trapezoid with lower base b1, upper base b2, and height h, area = ½ (b1 + b2) x h– b1 and b2 are parallel to each other

– The height h is a straight vertical line

Triangle

• No special formula for perimeter– Just add up the lengths of all the sides

• For a triangle with a height h and a base b, area = ½ b x h– The height h is a straight vertical line

Rectangle & Square

• Recall that a rectangle has two pairs of corresponding sides each equal in length

• Given a rectangle with length l and width w:– Perimeter = 2l + 2w– Area = l x w

• Recall that a square is a rectangle, but with four equal sides– i.e. l and w are the same length

• Given a square with a sides of length l:– Perimeter = 4l– Area = l2

Circle

• Recall that the line segment with endpoints at the center of the circle and on the outside of the circle is known as the radius

• Given a circle with radius r:– Circumference = 2πr

• Circumference is the circle’s equivalent to perimeter

– Area = πr2

π (pi) is a special constant in mathematics related to the circumference of a circle and its diameter• π is infinite so we often approximate it as 3.14

Perimeter & Area (Example)

Ex 1: Susan wishes to plant Black-Eyed Susans in her circular garden which has a radius of 5 feet. If a package of seeds will cover 12 square feet of her garden, how many whole packages of seeds must Susan buy in order to cover the entire garden?

Perimeter & Area (Example)

Ex 2: The owners of a 50 foot by 20 foot rectangular field have decided to install a walkway which will border the field. If the walkway extends 5 feet in all directions, find the perimeter of the walkway.

Perimeter & Area (Example)

Ex 3: Find the area of the shaded region:

a)

b)

Pythagorean Theorem

Pythagorean Theorem

• Pythagorean Theorem: given a right triangle with legs a & b and hypotenuse c, the following relationship exists: a2 + b2 = c2

– It does not matter which of the legs is a and which is b

– The hypotenuse, c, is the longest side AND is ALWAYS opposite the 90°-angle

• When solving problems with right triangles, it is often helpful to draw a picture

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Pythagorean Theorem (Example)

Ex 4: A 25-foot ladder leans on the roof of a 20 foot-tall building. How far from the building does the base of the ladder extend?

Pythagorean Theorem (Example)

Ex 5: Find the area of the parallelogram:

Summary

• After studying these slides, you should know how to do the following:– Solve problems involving area & perimeter of

common polygons and circles– Understand and apply the Pythagorean Theorem

• Additional Practice:– See problems in Section 10.3

• Next Lesson:– Volume & Surface Area (Section 10.4)