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# Unknown Dimensions of Triangles

## Use the formula A = (bh)/2 to solve for the unknown variable

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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 ft2\begin{align*}11.25 \ ft^2\end{align*} and a base length of 4.5 ft\begin{align*}4.5 \ ft\end{align*}, 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 m2\begin{align*}44 \ m^2\end{align*}. 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\begin{align*}h\end{align*}.

A4444÷1244(2)8811 m=12bh=128h=8h=8h=8h=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\begin{align*}h\end{align*}, 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.

AAAA=12bh=128(11)=12(88)=44 m2

We know from the problem that the area is 44 m2\begin{align*}44 \ m^2\end{align*}, 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=3inches\begin{align*}h = 3 inches\end{align*}

#### Example B

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

Solution: h=3feet\begin{align*}h = 3 feet\end{align*}

#### Example C

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

Solution: h=5meters\begin{align*}h = 5 meters\end{align*}

Here is the original problem once again.

If one of these triangles has an area of 11.25 ft2\begin{align*}11.25 \ ft^2\end{align*} and a base length of 4.5 ft\begin{align*}4.5 \ ft\end{align*}, what is the height of the triangle?

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

A=12bh\begin{align*}A = \frac{1}{2}bh\end{align*}

Now substitute in the given values.

11.25=124.5h\begin{align*}11.25 = \frac{1}{2}4.5h\end{align*}

11.25=2.25h\begin{align*}11.25 = 2.25h\end{align*}

11.25 div2.25=h\begin{align*}11.25 \ div 2.25 = h\end{align*}

h=5 ft\begin{align*}h = 5 \ ft\end{align*}

### 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\begin{align*}10.5 \ sq. in\end{align*} and a height of 6 in\begin{align*}6 \ in\end{align*}. 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=12bh\begin{align*}A = \frac{1}{2}bh\end{align*}

Now we fill in the given information.

10.5=12(6)b\begin{align*}10.5 = \frac{1}{2}(6)b\end{align*}

10.5=3b\begin{align*}10.5 = 3b\end{align*}

b=3.5\begin{align*}b = 3.5\end{align*}

The measure of the base of the triangle is 3.5 in\begin{align*}3.5 \ in\end{align*}.

### 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

### Vocabulary Language: English

Area

Area

Area is the space within the perimeter of a two-dimensional figure.
Base

Base

The side of a triangle parallel with the bottom edge of the paper or screen is commonly called the base. The base of an isosceles triangle is the non-congruent side in the triangle.
Height

Height

The height of a triangle is the perpendicular distance from the base of the triangle to the opposite vertex of the triangle.
Triangle

Triangle

A triangle is a polygon with three sides and three angles.