Calculating Perimeter and Area

Formulas are equations used to make specific calculations. Common formulas (equations) include:

P = 2l + 2w perimeter of a rectangle A = l + w area of a square or rectangle 2 E = mc Albert Einstein’s famous formula for calculating energy. 2

Calculating Perimeter Perimeter is the distance around the outside of a shape, or the sum of all the sides of any shape. Perimeter can be determined using grids, geoboards 1 3 and dot paper, and is calculated using equations called formulas.

Formulas for Perimeter 4 P = perimeter l = length s = side w = width

Shapes Formulas Illustration Words

P = s + s + s + s P = 6 + 6 + 6 + 6 = 24 Perimeter equals or units four times side. P = 4 × s 6 units P = 4s P = 4 × 6 = 24 units

P = s + s + s + s P = 7 + 3 + 7 + 3 = 20 Perimeter equals or units two times length 3 units P = 2 × l + 2 × w plus two times width. 7 units P = 2l + 2w P = 2 × 7 + 2 × 3 = 20 units

5 Perimeter equals 4 P = s + s + s P = 5 + 4 + 3 = 12 units side plus side plus side.

2 Perimeter equals 3 P = s + s + s + s + s P = 2 + 3 + 4 + 5 + 6 = 5 side plus side 20 units plus side plus 4 side plus side. 6

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The perimeter of a circle is called circumference. The formulas to calculate circumference are:

C = πd and C = 2πr

10 m C = πd = 3.14 × 10 m = 31.4 m

π = 3.14 c = circumference (the distance around the circle) r = radius (the distance from middle of circle to circumference) d = diameter (the distance across at the middle of the circle)

Calculating Area

Area is the space inside a flat, two-dimensional shape. For example, a carpet covers an area of the floor. The following words and abbreviations are used when talking about area:

A = area w = width l = length h = height b = base

The basic formula for calculating the area of 4-sided

shapes is length times width (l × w). Area is always 2 an amount “squared,” which is shown by writing the 4 m (squared) is: number 2 after and a little above the units. 1 m2 1 m2 units × units = units2

1 m2 1 m2 Example 2m × 2m = 4m2 1 m

1 m

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Formulas for Area of Quadrilaterals (four-sided shapes)

Shape Formula Example

A = 1 × w A = 3 × 3 = 9 units2

3 units

A = 1 × w A = 5 × 2 = 10 units2 2 units

5 units

A = b × h A = 12 × 10 = 120 units2 10 i

12 it

Area of Triangles If you fold a square or rectangle in half diagonally, you make two triangles. Try it!

The area of the triangle must be half of the area of the square.

The formula for calculating the area of a triangle is the A = b × h formula divided by 2.

× hb b = base A = 2 h = height

Shapes Formulas Illustration h × hb A = 4 × 6 6 units A = 2 = 24 ÷ 2 2 = 12 units 4 units b

×hb A = 2 × 3 A = 3 units 2 = 6 ÷ 2 = 3 units2 2 units

Mathematicians found that, when measured exactly, the circumference of a circle divided by its diameter always equals 3.14 (rounded off). You can prove this by wrapping a piece of string around a tubular object, measuring the string and then dividing that number by the diameter.

This constant repeated number is called pi (pronounced “pie”) and is represented by the symbol:

Working backwards, pi can be used to figure out the circumference of a circle—diameter multiplied by π or πd.

Shapes Formulas Illustration

2 r A = π r A = 3.14 × 4 × 4 • = 50.24 cm2 4 cm

A = π r2 r = d ÷ 2 = 12 ÷ 2 = 6 d A = 3.14 × 6 × 6 = 113.04 cm2 12 cm

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Practice: Calculating Perimeter and Area

1. Solve for perimeter in the following word problems.

a) Duane built a square patio in his back yard. The material he used came in 1 foot square pieces. How many pieces did Duane use to build his patio?

14 feet

b) Scott walked around the outside of his yard to decide where he should plant trees and hedges. How far did Scott walk?

Scott’s yard: 30 feet

25 feet

40 feet

15 feet

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2. Do the following activity to learn about circumference.

Materials: Objects to be measured: • measuring tape • juice can • soup can • string • coffee can • ruler or metre stick • garbage can top • fruit can

Directions: Use a measuring tape or the string and ruler/metre stick to measure the circumference of the tops of the objects. Then measure the length of the diameter.

List these measurements in a table like the one below:

Object Circumference Diameter Comparison 1) 2) 3) 4) 5)

a) How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large? This comparison is the ratio of the circumference to the diameter of the circle. This ratio is called π.

b) In the column marked COMPARISON, list the answer for the circumference divided by the diameter.

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3. Solve for the area of quadrilaterals, triangles and circles in these word problems.

a) Ariel added fertilizer to a circular flower bed that has a radius of 3.2 m. What is the area that Ari covered with fertilizer?

b) The glass on one side of the pyramid-shaped Muttart Conservatory in Edmonton was replaced. The height of the side is 100 ft. and the base is 56 ft. What is the area of the side that was replaced?

c) Leon needs to paint a cone shape on the wall of a gym. A diagram with dimensions is provided below. How much area will Leon cover with paint?

1.6 m

2.4 m

d) The swimming pool below needs a new winter cover. Calculate the area to be covered.

6 m

15 m

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4. Katya is an interior decorator. She has been hired to: • install carpet in all of the rooms of the house • put up a wallpaper border around the perimeter of each room.

Complete the chart to determine the amount of wallpaper border and carpet Katya must install.

Length of room Width of room Perimeter of Area of room room (for carpets) (for borders) 4 metres 8 metres 3 metres 9 metres 5 metres 25 metres2 10 metres 30 metres 25 metres 14 metres 11 metres 36 metres 24 metres 36 metres2 15 metres 225 metres2 28 metres 68 metres

Katya must install ______carpet. Katya must install ______wallpaper border.

5. Measure the length and width of rooms in your house/apartment or in a friend’s house. Use pencil and paper or a computer to draw the rooms. Rooms may include:

• living room • dining room • bathroom • laundry room • two bedrooms • kitchen

Label each room on the floor plan with the dimensions.

Calculate the perimeter and area of each room.

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