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# Unknown Dimensions of Triangles

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Unknown Dimensions of Triangles

Remember the triangles in the median of the Find Areas of Triangles Given Base and Height Concept? Take a look.

If one of these triangles has an area of $11.25 \ ft^2$ and a base length of $4.5 \ ft$ , what is the height of the triangle?

To figure this out, you will need to know how to use a formula and find the missing dimension of a triangle. This Concept will teach you how to do that.

### Guidance

Sometimes a problem will give us the area and ask us to find one of the dimensions of the triangle—either its base or its height. We simply put the information we know in for the appropriate variable in the formula and solve for the unknown measurement.

A triangle has an area of $44 \ m^2$ . The base of the triangle is 8 m. What is its height?

In this problem, we know the area and the base of the triangle. We put these numbers into the formula and solve for the height, $h$ .

$A & = \frac{1}{2} bh \\44 & = \frac{1}{2} 8h \\44 \div \frac{1}{2} & = 8h \\44(2) & = 8h \\88 & = 8h \\11 \ m & = h$

Remember, when you divide both sides by a fraction, you need to multiply by its reciprocal. To divide by one-half then, we multiply by 2. Keep this in mind when you use the area formula.

By solving for $h$ , we have found that the height of the triangle is 11 meters. Let’s check our calculation to be sure. We can check by putting the base and height into the formula and solving for area.

$A & = \frac{1}{2} bh \\A & = \frac{1}{2} 8(11) \\A & = \frac{1}{2} (88) \\A & = 44 \ m^2$

We know from the problem that the area is $44 \ m^2$ , so our calculation is correct.

Now try a few of these on your own.

Given the area and one other dimension, find the missing dimension of each triangle.

#### Example A

Base = 4 inches, Area = 6 sq. inches, what is the height?

Solution: $h = 3 inches$

#### Example B

Base = 5 feet, Area = 7.5 sq. feet, what is the height?

Solution: $h = 3 feet$

#### Example C

Base = 7 meters, Area = 17.5 sq. meters, what is the height?

Solution: $h = 5 meters$

Here is the original problem once again.

If one of these triangles has an area of $11.25 \ ft^2$ and a base length of $4.5 \ ft$ , what is the height of the triangle?

To figure this out, we can use the formula for finding the area of a triangle.

$A = \frac{1}{2}bh$

Now substitute in the given values.

$11.25 = \frac{1}{2}4.5h$

$11.25 = 2.25h$

$11.25 \ div 2.25 = h$

$h = 5 \ ft$

### Vocabulary

Triangle
a figure with three sides and three angles.
Area
the space enclosed inside a two-dimensional figure.
Base
the bottom part of the triangle.
Height
the length of the triangle from the base to the vertex.

### Guided Practice

Here is one for you to try on your own.

Find the missing base of the triangle.

A triangle has an area of $10.5 \ sq. in$ and a height of $6 \ in$ . What is the measure of the base?

To figure this out, we can solve for the base by using the formula for area of a triangle.

$A = \frac{1}{2}bh$

Now we fill in the given information.

$10.5 = \frac{1}{2}(6)b$

$10.5 = 3b$

$b = 3.5$

The measure of the base of the triangle is $3.5 \ in$ .

### Practice

Directions: Find the missing base or height given the area and one other dimension.

1. Area = 13.5 sq. meters, Base = 9 meters

2. Area = 21 sq. meters, Base = 7 meters

3. Area = 12 sq. meters, Base = 8 meters

4. Area = 33 sq. ft, Base = 11 feet

5. Area = 37.5 sq. ft. Base = 15 feet

6. Area = 60 sq. ft., height = 10 ft.

7. Area = 20.25 sq. in, height = 4.5 in

8. Area = 72 sq. in, height = 8 in

9. Area = 22.5 sq. feet, height = 5 feet

10. Area = 12 sq. in, height = 4 in

11. Area = 45 sq. in, height = 9 in

12. Area = 84 sq. ft, height = 12 ft

13. Area = 144 sq. in, height = 16 in

14. Area = 144.5 sq. ft, height = 17 ft.

15. Area = 123.5 sq. in, height = 19 in