Areas of Plane Figures

Find the missing measurements of each circle. Take 3.14 for . Give your answer to the nearest tenth. (Remember to include relevant unit or square unit in your answer) Radius = Diameter = Circumference = 62.8 cm Area = Radius = Diameter = Circumference = Area = 1962.5 square mm Radius = Diameter = Circumference = 37.68 cm Area = Radius = Diameter = Circumference = 14 m Area = Radius = Diameter = Circumference = Area = 0.64 square mm Radius = Diameter = Circumference = Area = 324 square mm Radius = Diameter = Circumference = 12 cm Area =

Radius = Diameter = Circumference = 18,84 mm Area =

Area of a Circle

Unit 2 > Lesson 3 of 6

The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter. We use the Greek letter (pronounced Pi) to represent the ratio of the circumference of a circle to the diameter. In the last lesson, we learned that the formula for circumference of a circle is: . For simplicity, we use = 3.14. We know from the last lesson that the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula: . The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2 However, it is easier to use one of the following formulas: or where is the area, and is the radius. Let's look at some examples involving the area of a circle. In each of the three examples below, we will use = 3.14 in our calculations.

Example 1: The radius of a circle is 3 inches. What is the area? Solution: = 3.14 (3 in) (3 in) = 3.14 (9 in2) = 28.26 in2

Example 2: The diameter of a circle is 8 centimeters. What is the area? Solution: 8 cm = 2 8 cm 2 = = 4 cm

= 3.14 (4 cm) (4 cm) = 50.24 cm2

Example 3: The area of a circle is 78.5 square meters. What is the radius? Solution: 78.5 m2 = 3.14 78.5 m2 3.14 = 25 m = =5m Summary: Given the radius or diameter of a circle, we can find its area. We can also find the radius (and diameter) of a circle given its area. The formulas for the diameter and area of a circle are listed below: or2

ExercisesDirections: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Use = 3.14 to calculate your answers. 1. The radius of a circle is 9 centimeters. What is the area? cm2CLEAR

ANSWER BOX: RESULTS BOX:

=

2. The diameter of a circle is 12 inches. What is the area? in2CLEAR

ANSWER BOX: RESULTS

=

BOX:

3. The radius of a circular rug is 4 feet. What is the area? ft2CLEAR


=

4. The area of a coin is 3.14 square centimeters. What is the radius?CLEAR


=

cm

5. The diameter of a bicycle wheel is 20 inches. What is the area of the wheel? in2CLEAR


=

Objective: I know how to calculate problems that involve the base, height and area of a triangle.

The formula for the area of triangle is Area = Given any two values of the formula, we can calculate the third value. Read the lessons on area of triangle if you need to learn how to calculate the area of a triangle.

Fill in all the gaps, then press "Check" to check your answers. Use the "Hint" button to get a free letter if an answer is giving you trouble. You can also click on the "[?]" button to get a clue. Note that you will lose points if you ask for hints or clues! Solve the following. (Remember to include the units or square units in your answer)

What is the area of a triangle with base 29.9 cm and height 25.6 cm?

What is the height of a triangle with base 24 mm and area 132 square mm?

What is the area of a triangle with base 11 m and height 23 m?

What is the length of the base of a triangle with height 12.1 m and area 117.975 square m?

What is the area of a triangle with base 20.3 mm and height 5.6 mm?


What is the length of the base of a triangle with height 18 m and area 99 square m?

What is the length of the base of a triangle with height 20 cm and area 150 square cm?


What is the length of the base of a triangle with height 16 mm and area 136 square mm?

Problem 1 What is the area of the triangle in the following picture?

Answer To find the area of the triangle on the left, substitute the base and the height into the formula for area. Area = (base)(height) Area = (3)(3) Area = 4.5 inches squared

Problem 2 Calculate the area of the triangle pictured below.

Answer To find the area of the triangle on the left, substitute the base and the height into the formula for area. Area = (base)(height) Area = (24)(27.6) Area = 331.2 inches squared Problem 3 Calculate the area of the triangle pictured below. Answer

Problem 4 Calculate the area of the triangle pictured below.

Answer

Problem 5 Calculate the area of the triangle pictured below. Answer

Problem 6 What is the area of the following triangle?

Answer

Problem 7 What is the area of the following triangle? Answer


Answer


Answer

The height is called the perpendicular height because it is at a right-angle to the base. Below is an example of the formula being used to find the area of a triangle.

Example QuestionBelow are two triangles, each with their base and perpendicular height indicated.

Practice Questions Work out the answer to each of these questions then click on the button marked to see whether you are correct.

What is the area of triangle A?

What is the area of triangle B?

ExercisesWork out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.

Question 1 Find the area of each of these triangles.

Part 1 (a) cm

(b)

m

(c)

cm

(d)

mm

(e)

cm

(f)

cm

(g)

cm

Word Problems: Area and Perimeter of TrianglesIn order to solve problems which require application of the area and perimeter for triangles, it is necessary to

Know how to use basic formulas for area and perimeter of a triangle. The area is A = bh where b is the length of the base of the triangle and h is thealtitude to that base. The perimeter is found by adding the lengths of the three sides. Solve basic linear and quadratic equations. Use the Pythagorean Theorem in finding the length, width, and hypotenuse of a right triangle. Know the formulas for the height and area of an equilateral triangle where s is the length of one side: and . Know basic trig functions and how to use them in a right triangle.

A typical problem involving the area and perimeter of a triangle gives us the area, perimeter and/or side lengths and altitude (we shall use the variable h for height) of the triangle. We may also be given a relationship between the area and perimeter or between the sides and altitude of the triangle. We may know one or more of the angles of the triangle. We need to calculate some of these quantities given information about the others. Two examples of this type of problem follow.

Example #1:

Suppose in a right triangle one of the legs is of length 5 and the angle formed by the hypotenuse and this leg is 28. What are the area andperimeter of this triangle? A diagram is shown below.

A right triangle is a special case of this type of problem since the legs are perpendicular. One leg is the base and the other is the height or altitudeof the triangle. In order to find h in the diagram above we use

h 2.6585

We can now find the area from

A = (1/2)bh = (1/2)(5)(2.6585) = 6.64625

To find the perimeter, we need the hypotenuse so we can add all three sides. We find the hypotenuse using the Pythagorean Theorem:

We add the lengths of the three sides and get

perimeter = 13.3213

Example #2:

Suppose a triangle with area 28 has a base of 7x and a height of 4x as shown in the diagram below. What are the area, the perimeter, and the dimensions of this triangle?

We know area is 28 and equals bh. We write

Next we solve the equation 28 = 14x2 for x

x=

We now know that

We can check our work by recalculating the area

A=

The frustrating part of this analysis is that we cannot find the measures of the other two sides with the information that we have been given. Therefore, we can solve for x, and get the lengths of the altitude and base, but we will never be able to determine either the dimensions or theperimeter of this triangle.

Let's Practice

Question #1 The altitude of a right triangle is twice the base. If the area is 36, what are the dimensions and the perimeter of this triangle?

Question #2 A equilateral triangle has a base of 7. What are the altitude, the area and the perimeter of the triangle?

Question #3 A right triangle has a hypotenuse of 10 and one acute angle of 42. Find the area and perimeter of this triangle.

Try These Question #1 A triangle has a base of b = 20 cm. If the height equals 2x cm and its area equals 80x2 cm2, find the values of its height and area. A. h = 1 cm and A = 20 cm2

B. h = 1/4 and A = 5 cm2 C. h = 1/2 and A = 5 cm2

Question #2 An equilateral triangle has an altitude of 6 inches. What are the perimeter and area of this triangle? A. perimeter = inches and area = square inches

B. perimeter = C. perimeter = square inches

inches and area = inches and area =

square inches

Question #3 A right triangle has one leg of length 7 which makes an angle of 36 with the hypotenuse. What are the area and perimeter of thistriangle correct to one decimal place? A. area = 14.4 and perimeter = 19.2 B. area = 17.8 and perimeter = 16.9 C. area = 17.8 and perimeter = 20.7

Area of a Triangle

Unit 1 > Lesson 4 of 8

The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional like a carpet or an area rug. A triangle is a threesided polygon. We will look at several types of triangles in this lesson.

To find the area of a triangle, multiply the base by the height, and then divide by 2. The division by 2 comes from the fact that a parallelogramcan be divided into 2 triangles. For example, in the diagram to the left, the area of each triangle is equal to one-half the area of the parallelogram.

Since the area of a parallelogram is , the area of a triangle must be one-half the area of a parallelogram. Thus, the formula for the area of a triangle is: or where is the base, is the height and means multiply.

The base and height of a triangle must be perpendicular to each other. In each of the examples below, the base is a side of the triangle. However, depending on the triangle, the height may or may not be a side of the triangle. For example, in the right triangle in Example 2, the height is a side of the triangle since it is perpendicular to the base. In the triangles in Examples 1 and 3, the lateral sides are not perpendicular to the base, so a dotted line is drawn to represent the height.

Example 1: Find the area of an acute triangle with a base of 15 inches and a height of 4 inches. Solution:

=

(15 in) (4 in) (60 in2)

=

= 30 in2

Example 2: Find the area of a right triangle with a base of 6 centimeters and a height of 9 centimeters. Solution:

=

(6 cm) (9 cm) (54 cm2)

=

= 27 cm2

Example 3: Find the area of an obtuse triangle with a base of 5 inches and a height of 8 inches. Solution:

=

(5 in) (8 in) (40 in2)

=

= 20 in2

Example 4: The area of a triangular-shaped mat is 18 square feet and the base is 3 feet. Find the height. (Note: The triangle in the illustration to the right is NOT drawn to scale.) Solution: In this example, we are given the area of a triangle and one dimension, and we are asked to work backwards to find the other dimension.

18 ft2 =

(3 ft)

Multiplying both sides of the equation by 2, we get: 36 ft2 = (3 ft)

Dividing both sides of the equation by 3 ft, we get: 12 ft = Commuting this equation, we get: = 12 ft

Summary:

Given the base and the height of a triangle, we can find the area. Given the area and either the base or the height of a triangle, we can find the other dimension. The formula for area of a triangle is: or where is the base, the height is

ExercisesDirections: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. Your answers should be given as whole numbers greater than zero. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

1. Find the area of a triangle with a base of 16 feet and a height of 3 feet.


=

ft2

CLEAR

2. Find the area of a triangle with a base of 4 meters and a height of 14 meters.


=

m2

CLEAR

3. Find the area of a triangle with a base of 18 inches and a height of 2 inches.


=

in2

CLEAR

4. A triangular-shaped piece of paper has an area of 36 square centimeters and a base of 6 centimeters. Find the height. (Hint: work backwards) ANSWER BOX: RESULTS BOX: = cmCLEAR

5. The area of a triangular-shaped rug is 12 square yards and the height is 3 yards. Find the base. (Hint: work backwards) ANSWER BOX: =

Areas of Plane Figures

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