# Area of a Triangle

The** area of a triangle** A is half the product of its base b and its height h. The height of a triangle is also known as the altitude. This formula works only if the base is perpendicular to the height. That means the base and the height form a right angle.

## Examples Involving Area of Triangle Formula

**Example 1:** Find the area of a triangle with a base of 5 and a height of 3.

We have \color{blue}b=5 and \color{red}h=3. Plug the values into the formula then simplify.

Since the unit of measure is not specified, we can write the final answer as A = 7.5 square units.

**Example 2:** Find the area of triangle ABC.

Clearly, the base is \color{blue}9 feet while the height is \color{red}4 feet. Observe that the base and height meet at a 90-degree angle. To find the area of the given triangle, we multiply the base and height then divide the product by 2.

Notice that the height of the triangle is found inside of the triangle.

The final answer is 18 square feet.

**Example 3:** Draw a right triangle with legs of length 5 inches and 7 inches. Calculate its area.

The legs of a right triangle are the sides that form a right angle, {90^ \circ }. More so, the legs also serve as the height and base of the triangle. If we let the base be the shorter leg, this forces the longer leg to be the height. Thus, we have \color{blue}b=5 and \color{red}h=7.

Notice that the height is part of the triangle itself, which is one of the two legs. In fact, any of the legs can be designated as height since they intersect at a 90-degree angle.

Calculating the area of the right triangle, we have

The final answer is 17.5 square inches.

**Example 4:** Sketch an obtuse triangle with a base of 4 centimeters and an exterior height of 4.5 centimeters. Then, find its area.

An obtuse triangle is a type of triangle wherein one of its interior angles has a measure of greater than 90 degrees but less than 180 degrees.

In this scenario, the height of the triangle is formed outside, and to be exact, opposite to one of the acute angles. The height is perpendicular to the extended base.

Since \color{blue}b=4 and \color{red}h=4.5, the area of the triangle is computed as follows.

Therefore, the final answer is 9 square centimeters.

**Example 5:** What is the area of a triangle whose base is 2.5 feet and height is 12 inches?

Before we can apply the formula to find the area, we need to ensure that the units of measure are the same. We can either express the area in terms of square feet ft^2 or square inches in^2.

To express the area in **square feet**, we will keep the base of 2.5 feet but convert the height of 12 inches into feet. By conversion, 12 inches is equal to 1 foot. Now, we have \color{blue}b=2.5 ft and \color{red}h=1 ft.

In the same manner, to express the area in **square inches**, we would keep the height of 12 inches but convert 2.5 feet into inches. Since 1 foot is equal to 12 inches, that means 2.5 feet equals 30 inches, 2.5 \times 12 = 30. Finally, we have \color{blue}b=30 in. and \color{red}h=12 in.

You can give your answer in either in^2 or ft^2. It is both correct since

1.25\,f{t^2} = 180\,\,i{n^2}.

**Example 6:** The base of a triangle is 17 meters. Its area is 204 square meters. Find the height of the triangle.

This is a very straightforward problem. We substitute the known values into the formula then solve for the leftover variable.

In this case, we know the **base** b and the **area** A of the triangle. That means we can solve for the height h.

Therefore, the height of the triangle is 24 meters.

**Example 7:** The area of a triangle is 162.56 square centimeters. What is the base of the triangle in **inches** if the height is 16 centimeters?

This problem is a combination of examples #5 and #6. After we find the base in centimeters, we will convert it into inches using the conversion factor 1 in. = 2.54 cm.

Since the area of the triangle is 162.56 *cm *and its height is 16* cm*, we have

The base is 20.32 centimeters. However, we still need to convert centimeters to inches to get to our final answer.

Since 1 in. = 2.54 cm, we divide 20.32 by 2.54 to get to inches.

{\Large{{{20.32\,} \over {2.54}}}} = 8

Therefore, the base is 8 inches long.

**Example 8:** The base of a triangle is twice its height. If the area is 289 square kilometers, find the measure of the base.

It is given that the base is twice its twice, {\color{blue}b}=2{\color{red}h}. So if the height is h then the base is 2h.

height = \color{red}h

base = 2{\color{red}h}

The area is also given as 289 square kilometers.

Area = 289

Plugging these values into the formula and solving for \color{red}h.

Since the base is 2\color{red}h, then

Therefore, the base is 34 km.

Let’s check if our base and height are correct. For a triangle with a base of 34 km and a height of 17 km, the area must be 289 square km.

Yes, it checks. That base and height give us an area of 289 km^2.

**Example 9:** The base of the triangle is 5 units longer than its height. If the area of the triangle is 117 square units, what is the measure of the height?

Let h be the height of a triangle. The base is said to be 5 units longer than the height which can be expressed as b=h+5.

Since b=h+5 and the area of a triangle is 117 square units, we can plug this information into the formula to get a quadratic equation solvable using the Factoring Method.

We will ignore the negative solution because there’s no such thing as a negative height.

Therefore, the **height** is 13 units.

We can perform a quick check to verify if our answer is correct.

If h=13, the base is 13+5=18. Calculating the area of the triangle we get

Yes, we get the same area of 117 square units. That means our values for the base and height are correct!

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