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Calculating the Area of Triangles
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Skills Application: Calculating the Area of Triangles
Step-by-Step: Calculating the Area of Triangles
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Triangles are frequently encountered in construction because triangles are one of the strongest shapes. Triangles are found in scaffolding, roof trusses and the frames of tower cranes. The engineering principle is simple: the bigger the area of a triangle, the more rigid the structures are.

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Area of Triangles

The area of a triangle with base B and height H is one-half of its base times its height:

A = B × H ÷ 2

EXAMPLE:

The area of a triangle with base 5 centimeters and height 4 centimeters equals:

A = 5 × 4 ÷ 2 = 10 cm²

The formula then looks like this:

B x A ÷ 2

The problem with altitude (A) is that the word “area” is also abbreviated as (A)

The full formula then could look like:

A = B x A ÷ 2

To avoid confusion, the word altitude is not used on this site, only in aviation context.

Check out the ADVANCED tab. The exercises will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, to sketch the shapes, label the shapes with the given numbers, tracking changes to units of measure, visualizing to cross-link the concepts of area and numbers. We use the numbers from this gable stuccoing example. If you spend 20 – 40 min working with the different ideas in the ADVANCED page, youll remember key formulas easier and youll learn new skills faster.

### A Short Review

Area of a triangle is calculated by multiplying its Base by half its Height:

$A = B \cdot\frac{H}{2}$

Depending on where a triangle is, the words Base and Height may be replaced by 2 other measurements, such as Run and Rise in roofs, Run and Drop with drainage and roads, Distance and Altitude in site layout and surveying.

Regardless of the words used, the math stays the same: multiply 2 numbers and divide by 2 to get Area.

Triangles can form parts of complex shapes: unequal pitch intersecting hip roofs, gem stone facets, straight and curved stairs, coil springs, screw threads, cones, hoppers, feeders all are based on flat or curved triangles. Triangular prisms have triangular bases, and pyramids with any bases have a number of triangles that make up the sides. Look for triangles in objects.

### Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

### Worksheet: Level 1 Sample Questions

Match the numbers in the picture with the corresponding dimensions on the right and fill in the amounts:

Example:

Further examples:

Does the triangle Area formula only work for:

1. right angle triangles?
2. isosceles triangles?
3. any triangle?

### Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

### Worksheet: Level 2 Sample Questions

Calculate the area of the following triangles. Answer in decimals.

Example:

Base = 30, height = 6, Area = ?

${A = }\frac{b \cdot h}{2}$

base × height ÷ 2 = area

30 × 2 ÷ 6 = 10

Question:

1. b = 5, h = 2, A = ?

Further examples:

Calculate the Area of a triangle with a Base of 102 cm and a Height of 27 cm.

### Worksheet: Level 2 Sample Answer Key

${A =}\frac{b \cdot h}{2}$

base × height ÷ 2 = area

5 × 2 ÷ 2 = 5

### Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

### Worksheet: Level 3 Sample Questions

Remember:

${A = }\frac{b \cdot h}{2}$

base × height ÷ 2 = area

Question:

1. What is the area of a triangular planter if its shortest side walls are 11 m and 14 m?

Further examples:

Calculate the Area of one side in the Great Pyramid of Giza with a Length of 756 feet and a Peak Height of 455 feet.

### Worksheet: Level 3 Sample Answer Key

Note: It does not matter which number is base and which is height.

${A =}\frac{b \cdot h}{2}$

base × height ÷ 2 = area

11 ? × 14 ? ÷ 2 = 77 m²

Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

One Way to Get the Answer

Calculate the area of a triangle, base = 30 feet, height = 6 feet

One way to get the answer, using the formula

$\frac{b \cdot h}{2}{= A}$

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this:

Using a calculator, do the following:

Enter 30

Press x (multiplication) button

Enter 6

Press ÷ button

Enter 2

Press =

You should see 90 on your display.

Get a \$20-or-so dual-display scientific calculator, even if you cant take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you cant do without

Another way to get the answer

Calculate the area of a triangle, base = 30 feet, height = 6 feet

Another way to get the answer, using the formula

$\text{A = }\frac{b}{2}\cdot\text{h}$

1.  Identify that the area of a triangle is to be calculated.

2.  Make a sketch of the triangle:

3. Label the triangle. Use either full words or abbreviate.

The idea is to visualize the problem before any math is done.

4. Write the formula

$\text{A = }\frac{b}{2}\cdot\text{h}$

5.  Re-write the formula: change all of the letters into words, according to their meaning:

base ÷ 2 x height  = area

6.  Re-write the formula: change all of the words you can for actual given numbers:

Leave out units of measure words. (eg. feet, cm)

30 ÷  2 x  6 = area

7.  Calculate: 30  ÷ 2 x     6        =  90

8.  Take a pencil and write down  “area = 90” or “A = 90”

9.  Look at the units of measure. ft was multiplied with ft. The unit for area is ft2.

Write ft2 after the 90 Now youre done with the math.

10.  The last step: check your work, make sure everything was copied and written correctly then determine that the correct way to write the answer is 90 ft2

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this:

Using a calculator, do the following:

Enter 30

Press ÷ button

Enter 2

Press x button (multiplication)

Enter 6

Press =

You should see 90 on your display.

Another way to get the answer

Calculate the area of a triangle, base = 30 feet, height = 6 feet

Another way to get the answer, using the formula:

$\text{A = }\frac{h}{2}\cdot\text{b}$

1.   Identify that the area of a triangle is to be calculated.

2.   Make a sketch of the triangle:

3.   Label the triangle. Use either full words or abbreviate.

The idea is to visualize the problem before any math is done

4.   Write the formula:

$\frac{h}{2}\cdot\text{b = A}$

5.   Re-write the formula: change all of the letters into words, according to their meaning:

height  ÷ 2 x base = area

6.   Re-write the formula: change all of the words you can for actual given numbers:

Leave out units of measure words. (eg. feet, cm)

6 ÷  2 x 30  = area

7.   Calculate  6  ÷ 2 x  30  =  90

8.   Take a pencil and write down  “area = 90” or “A = 90”

9.      Look at the units of measure. ft was multiplied with ft. The unit for area is ft²

Write ft² after the 90 Now youre done with the math.

10.  The last step: check your work, make sure everything was copied and written correctly then determine that the correct way to write the answer is 90 ft²

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this:

Using a calculator, do the following:

Enter 6

Press ÷ button

Enter 2

Press x (multiplication)

Enter 30

Press =

You should see 90 on your display.

Compare the 3 area calculations. The results are the same, 90 ft2. Both methods have exactly 2 calculations. The formulas look different. But they are really not.

The first method

$\frac{b \cdot h}{2}$

has a multiplication, then a division. Since multiplication is interchangeable it could have been written as

$\frac{h \cdot b}{2}$

Try it. Do you get 90?

The second method

$\frac{b}{2}\cdot {h}$

has a division up front, then a multiplication. Since multiplication is interchangeable it could also be written as

${h}\cdot\frac{b}{2}$

Try it. Do you get 90?

The third method

$\frac{h}{2}\cdot{b}$

has a division up front, then a multiplication. Since multiplication is interchangeable it could also be written as

${b}\cdot\frac{h}{2}$

Try it. Do you get 90?

So now we have 6 possible ways to calculate the area of triangles:

${A =}\frac{b \cdot h}{2}$

${A =}\frac{h \cdot b}{2}$

${A =}\frac{b}{2}\cdot{h}$

${A = h}\cdot\frac{b}{2}$

${A = b}\cdot\frac{h}{2}$

${A =}\frac{h}{2}\cdot{b}$

Same results, same number of steps, same amount of work. A formula is a symbolic short-hand way to write the above 2-step procedures. Use a formula that`s easiest to remember for you.