Perimeters of Polygon

• Perimeter is the sum of the lengths of its sides.Theorem:

The perimeter P of a triangle with sides a, b, and c is given by

P= a+b+cEx.

A triangular piece of paper measures 8 cm, 12 cm, and 15 cm. What is the peri-meter of the piece of paper?

Solution:P= a+b+c = 8 cm+ 12 cm+ 15cm = 35 cm

Perimeter of a rectangle- is the sum of twice its length and twice its width

Theorem:The perimeter of a rectangle is the sum of

twice its length and twice its width.P= 2l + 2w

• Ex. A rectangular table cloth

has a width of 1.5 m and a length of 2.8 m. How many meters of lace trimmingsshould be bought to makeits borders?Solution:

P= 2l+ 2w = 2(2.8 m) + 2(1.5 m)

= 8.6 m

Perimeter of a SquareTheorem:The perimeter P of a square with side s is given by

P= 4sEx.

One side of a square matIs of length 45.5 cm. How long of a lace material is needed to put borders on it?

Solution:P= 4s = 4(45.5 cm) = 182 cm

45.5 cm

Finding the area of a reg. polygon?

Suppose reg. n-agon w/ side s

The radii divides the figure into n isos Δs each w/ area = ½ as Apothem Side

• Area of n-gon = n • ½ as• Perimeter of n-gon = ns

s

p = nsPerimeter of reg. Polygon # of sides Length of

sides

The area formula for any reg. Polygon

A = ½ ap

Area of any reg. Polygon

ApothemPerimeter of reg. polygon

Example: Find the area of a regular decagon with a 12.3in apothem and an 8in side.

A = ½ ap

= ½ (12.3in)(80in)

= 492in2

p = ns

= (10)(8cm)

= 80cm

Example: Find the apothem of a reg. hexagon with sides of 10mm.

10mm

12

3

m1 = 360/6 = 60

m2 = ½ (60) = 30

m3 = 60

a

30

605mm

30-60-90 Δ shortcut

a = √3 • short side

a = √3 • (5mm)

a = 5√3 = 8.66

Could you find the area of it now??

Example: Find the area of a regular pentagon with 7.2ft sides and a 6.1ft radius.

6.1ft 6.1fta

A = ½ ap

= ½ (4.92ft)(36ft)

= 88.6ft2

3.6 ft

p = ns

= (5)(7.2ft)

= 36ft

a2 + b2 = c2

a2 + 3.62 = 6.12

a2 + 12.96 = 37.21

a2 = 24.25

a = 4.92

What is the difference between the radius and the apothem?

What is the formula for the area of a regular polygon?

A = ½asn A = ½aP

The radius is longer because it is from the center all the way to the edge of the circle but the apothem is just from the center tothe middle of the side

Circumference of a Circle• Theorem:

The circumference C of a circle with a diameter d or radius r is given by

C= πd or C= 2πrEx.

The diameter of a five-pesoCoin is 2.7 cm. Find its circumference.

Solution:C= πd = π(2.7 cm) = 2.7π cm

Sample Problems:1. Find the circumference of a circular table

whose diameter is 60.48 cm.2. If the length and the width of the floor of a

studio-type room are 5m and 4m, respectively, what is the perimeter of that room?

3. A square garden is to be fenced. One side is 5 ¾ m. How long is the fence needed to surround it on all sides?

4. Find the circumference of a hula hoop if its diameter is 1.1 m.

5. Find the distance around a triangle whose sides are 12 ½ cm, 15 cm, and 9 cm.

Area of a RectangleTheorem:

The area A of a rectangle of length l and width w is given by

A= l x wEx.

The glass top of a table has a length of 105 cm and a width of 61 cm. What is its area?

Solution:A= l x w = 105 cm x 61 cm = 6405 cm2

Area of a SquareTheorem:

the area A of a square of side is given byA= s2

Ex.What is the floor area of a square room which measures 5.5 m on each of its sides?

Solution:A = s2

=(5.5 m)2

= 30.25 m2

Areas of a TriangleTheorem:

The area A of a triangle with base b and height h is given byA= ½ bh

Ex.The base of a triangular flaglet is 6 cm long. If the height of the flaglet is 3.2 cm, what is its area?

Solution:A = ½ bh

= ½ (6 cm x 3.2 cm)= 9.6 cm2

Area of a ParallelogramTheorem:

The area A of a parallelogram with base b and height h is given byA= bh

Ex. A ricefield is in the shape of a parallelogram. If its

base is 38 m and its height is 25 m, what is its area?Solution:A = bh

= (38 m)(25 m)= 950 m2

Area of a TrapezoidTheorem:

The area A of a trapezoid of height h and bases b1 and b2 is given by

A= ½ h(b1 + b2)Ex.

Find the area of a trapezoid with a height of 8 cm and bases of 10 cm and 5 cm.

A = ½ h(b1+b2)= ½ (8 cm)(10 cm + 5 cm)= (4 cm) (15 cm)= 60 cm2

Area of a CircleTheorem:

The area A of a circle of radius r is given by

A = πr2

Ex. Find the area of a circle with a diameter of 6 cm.

Solution:A = πr2

= π (3 cm)2

= 9π cm2

Solve the following:1.The lower base of a trapezoid is twice as long as the

upper base, If the height of the trapezoid is 2 cm and its area is 12 cm2, what are the lengths of its bases?

2. Find the area of a circle whose radius is 0.82 meter.3. The area of a rectangular swimming pool is 375

square meters. If the length is 25 meters, what is its width?

4. Each side of a marble tile is 16 cm long. How many tiles are needed to cover an area of 5120 square centimeters?

5. A triangle has an area of 45 cm2 and a base of 5 cm. What height corresponds to this base?