Area of a Right Triangle - Math Steps, Examples & Questions

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Area of a right triangle

Here you will learn how to find the area of a right triangle, including what formula to use and why it works.

Students will first learn about the area of right triangles as part of geometry in 6th grade.

What is the area of a right triangle?

The area of a right triangle is the amount of the space inside a right triangle.

A right triangle is a type of triangle that has a right angle. The right angle is always opposite of the longest side of the triangle.

In order to find the area of a right triangle, start with the area of a rectangle.

The area of the rectangle is calculated by multiplying the $\text { base } \times \text { height}.$

Within a rectangle, there are two congruent right triangles. They can be shown by connecting two opposite vertices.

The area of the rectangle is the same as two congruent right triangles. This means the area of each right triangle is exactly half the area of the rectangle.

The general formula to find the area of any triangle is:

$\text { Area of a triangle } = \cfrac{\text { base } \times \text { height }}{2}$

This can also be written as the following formula:

$A=\cfrac{1}{2} \, b h$

where $b$ is the base length and $h$ is the height of the triangle.

The space inside of shapes is measured in square units, so the final answer must be given in $\text {units}^2$ . For example, $\mathrm{cm}^2, \mathrm{~m}^2, \mathrm{~mm}^2$ .

For example,

What is the area of the right triangle?

$A=\cfrac{1}{2} \, b h$

The base is $4$ and the height is $3$ .

A few things to note:

The height of a triangle is always a perpendicular line that starts at a vertex and intersects with the opposite side of the triangle at a $90$ degree angle.
The other side lengths of the triangle may be given, but they are extra information.

$\begin{aligned} & A=\cfrac{1}{2} \times 4 \times 3 \\\\ & A=2 \times 3 \\\\ & A=6 \end{aligned}$

The area of the right triangle is $6 ft^2$ .

What is area of a right triangle?

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Common Core State Standards

How does this relate to 6th grade math?

Grade 6 – Geometry (6.G.A.1)
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

How to find the area of a right triangle

In order to find the area of a right triangle:

Identify the base and perpendicular height of the triangle.
Write the area formula.
Substitute known values into the area formula.
Solve the equation.
Write the answer, including the units.

Area of right triangle examples

Example 1: given base length and height

Find the area of the triangle below:

Identify the base and perpendicular height of the triangle.

$b = 8 \, cm$

$h = 7 \, cm$

2Write the area formula.

$A=\cfrac{1}{2} \, b h$

3Substitute known values into the area formula.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (8)(7) \end{aligned}$

4Solve the equation.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (8)(7)\\\\ &=28 \end{aligned}$

5Write the answer, including the units.

$A=28 \mathrm{~cm}^2$

Remember: Your final answer must be in units squared.

Example 2: given base length and height

Find the area of the triangle below:

Identify the base and perpendicular height of the triangle.

$b = 6 \, m$

$h = 200 \, cm$

Notice, there are $2$ different units here. You must convert them to a common unit:

$200 \, cm = 2 \, m$

$b = 6 \, m$

$h = 2 \, m$

Write the area formula.

$A=\cfrac{1}{2} \, b h$

Substitute known values into the area formula.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (6)(2) \end{aligned}$

Solve the equation.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (6)(2)\\\\ &=6\end{aligned}$

Write the answer, including the units.

$A=6 \, m^2$

Example 3: given base length and height and hypotenuse

Find the area of triangle $BAC$ below:

Identify the base and perpendicular height of the triangle.

$b = 12 \, cm$

$h = 5 \, cm$

Note: The $13 \, cm$ is not needed to solve. It is extra information.

Write the area formula.

$A=\cfrac{1}{2} \, b h$

Substitute known values into the area formula.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (12)(5) \end{aligned}$

Solve the equation.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (5)(12) \\\\ &=30 \end{aligned}$

Write the answer, including the units.

$A=30 \, cm^2$

Example 4: area of the isosceles right triangle – fraction

Find the area of triangle below:

Identify the base and perpendicular height of the triangle.

Rotating the triangle makes it easier to identify the base and height.

$\begin{aligned} & \mathrm{b}=2 \, \cfrac{1}{2} \\\\ & \mathrm{h}=2 \, \cfrac{1}{2} \end{aligned}$

Write the area formula.

$A=\cfrac{1}{2} \, b h$

Substitute known values into the area formula.

$\begin{aligned} A & = \cfrac{1}{2} \, b h \\\\ & = \cfrac{1}{2} \, \left(2 \, \cfrac{1}{2} \, \right) \left(2 \, \cfrac{1}{2} \, \right) \end{aligned}$

Solve the equation.

$\begin{aligned} A & =\cfrac{1}{2} \, b h \\\\ & =\cfrac{1}{2} \, \left(2 \, \cfrac{1}{2} \, \right) \left(2 \, \cfrac{1}{2} \, \right) \\\\ & =3 \, \cfrac{1}{8} \end{aligned}$

Write the answer, including the units.

$A=3 \, \cfrac{1}{8} \text { units}^2$

Note, when no units are given the answer can be labeled with the general term $\text {units}^2.$

Example 5: compound shape

Below is the floor plan for a new deck that needs to be painted. One can of paint costs $\$12$ and covers an area of $2.5 \, m^2.$ How much would it cost to paint the entire deck?

Identify the base and perpendicular height of the triangle.

Split the plan into $2$ shapes; a rectangle and a right triangle.

For the triangle:

$b = 3.5 \, m$

$h = 2.5 \, m$

Write the area formula.

$A=\cfrac{1}{2} \, b h$

Substitute known values into the area formula.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (3.5)(2.5) \end{aligned}$

Solve the equation.

$\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (2.5)(3.5)\\\\ &=4.375 \end{aligned}$

Now you must find the area of the rectangle.

$\begin{aligned} \text { Area of a rectangle }&=l \times w \\\\ &=2.5 \times 3 \\\\ &=7.5 \end{aligned}$

Write the answer, including the units.

Total Area $= 4.375+7.5=11.875 \, m^2$

Now divide $11.875$ by $2.5$ since one can of paint only covers an area of $2.5 \, m^2$ .

$11.875 \div 2.5=4.75$

Since you can’t buy $0.75$ paint cans and $4.75$ is not quite enough, you have to buy $5$ cans of paint.

Multiply to find the cost: $5 \text { cans of paint } \times \$ 12=\$ 60$ .

The total cost to paint the deck is $\$60$ .

Example 6: calculating base length

Triangle $XYZ$ is a right triangle with an area of $15 \, cm^2$ . The height of the triangle is $5 \, cm$ . Find the base length of the triangle.

Identify the base and perpendicular height of the triangle.

$\begin{aligned} A &=15 \mathrm{~cm}^2 \\\\ b &= \, ? \\\\ h &=5 \mathrm{~cm} \end{aligned}$

Write the area formula.

$A=\cfrac{1}{2} \, b h$

Substitute known values into the area formula.

$\begin{aligned} & A=\cfrac{1}{2} \, b h \\\\ & 15=\cfrac{1}{2} \, \times b \times 5 \\\\ & 15=\cfrac{1}{2} \, \times 5 \times b \\\\ & 15=2.5 \times b \end{aligned}$

Solve the equation.

$15=2.5 \times b$

To solve, find what times $2.5$ is equal to $15$ . Since $2.5 \times 6=15$ , the base is $6$ .

Write the answer, including the units.

$b=6 \mathrm{~cm}$

The units are not squared, because the base is the length of the bottom of the triangle (one-dimensional), not the square units within the triangle (two-dimensional).

Teaching tips for the area of a right triangle

Start by connecting the area of right triangles to the area of rectangles. One easy way to do this is to have students take a rectangular piece of paper and cut it diagonally – forming $2$ right triangles. Then ask students to make conjectures about the area of the triangles – including discussing their ideas with other students and supporting any claims with evidence and mathematical reasoning.

Choose worksheets that include a variety of right triangles, in all orientations. This includes right triangles with the right angle on the right side, left side or at the top of a triangle (with the hypotenuse appearing horizontally).

This forces students to think flexibly about the position of the triangle and requires them to think about the perpendicular relationship between the base and the height, instead of just assuming that the two measurements given are what is needed to calculate the area.

Easy mistakes to make

Confusing which measures represent the base and the height
A question may give more than two measurements on a triangle, some of which will not be needed. Carefully identify the perpendicular relationship between the base and the height to use the correct measurements.
For example,
To calculate the area here, you only need the base and height.

$b = 4 \, cm$
$h = 3 \, cm$
$\text {Area }=\cfrac{1}{2} \, \times 4 \times 3=6 \mathrm{~cm}^2$
You can ignore the length of the hypotenuse $(5 \,cm)$ – it is not needed to solve for the area.

Thinking the sides of a right triangle can’t be equal
While a right triangle cannot be an equilateral triangle, it can be an isosceles. The two legs that form the right angle can be equal.
For example,

Forgetting to record the units
It is common to forget the units for area in the final answer. When calculating area, your answer must always have square units.

Practice area of right triangles questions

108 \, cm^{2}

54 \, cm^{2}

27 \, cm^{2}

36 \, cm^{2}

The area of a triangle is calculated with the formula: $A=\cfrac{1}{2} \, b h$ .

Rotating the triangle makes it easier to identify the base and height.

Area of a Right Triangle image 21 US

For this triangle:

$b = 12 \, cm$

$h = 9 \, m$

Substitute the values into the formula.

$\begin{aligned} A & =\cfrac{1}{2} \, b h \\\\ & =\cfrac{1}{2} \, (12)(9) \\\\ & =54 \mathrm{~cm}^2 \end{aligned}$

2,250 \, cm^{2}

4,500 \, cm^{2}

45 \, cm^{2}

22.5 \, cm^{2}

The area of a triangle is calculated with the formula: $A=\cfrac{1}{2} \, b h$ .

Area of a Right Triangle image 23 US

For this triangle:

$b = 0.9 \, m$

$h = 50 \, cm$

Since the measurements are in different units $(m$ and $cm),$ they need to be converted to the same unit:

$0.9 \, m = 90 \, cm.$

$b = 90 \, cm$

$h = 50 \, cm$

Note: $103 \, cm$ is not needed to solve. It is extra information.

Substitute the values into the formula.

$\begin{aligned} A & =\cfrac{1}{2} \, b h \\\\ & =\cfrac{1}{2} \, (90)(50) \\\\ & =2,250 \mathrm{~cm}^2 \end{aligned}$

$2$ meerkats

$3$ meerkats

$24$ meerkats

$25$ meerkats

The shape can be split into a rectangle and a triangle. The opposite sides of the rectangle are congruent.

Area of a Right Triangle image 25 US

The area of the rectangle is $4.5 \times 3.2 = 14.4 \, m^{2}$

The area of the triangle is $\cfrac{1}{2} \, \times 4.5 \times 4.5 = 10.125 \, m^{2}$

This means the total area is $14.4+ 10.125 = 24.525 \, m^{2}$

Now to calculate how many meerkats will fit into the enclosure, take $24.525 \, m^2$ and divide it by $9 \, m^2,$ because each meerkat needs that much space to roam.

$24.525 \div 9 = 2.725$

The quotient has $2$ wholes and $725$ thousandths.

There cannot be part of a meerkat and $2.725$ is not enough for $3$ meerkats.

So, $2$ meerkats will fit in the enclosure.

54 \, cm^{2}

101.82 \, cm^{2}

72 \, cm^{2}

51 \, cm^{2}

The area of a triangle is calculated with the formula:

$A=\cfrac{1}{2} \, b h$ .

Rotating the triangle makes it easier to identify the base and height.

Area of a Right Triangle image 27 US

For this triangle:

$b = 6 \, cm$

$h = 17 \, cm$

Substitute the values into the formula.

$\begin{aligned} A & =\cfrac{1}{2} \, b h \\\\ & =\cfrac{1}{2} \, (6)(17) \\\\ & =51 \mathrm{~cm}^2 \end{aligned}$

27.5 \mathrm{~ft}^2

55 \mathrm{~ft}^2

22 \mathrm{~ft}^2

44 \mathrm{~ft}^2

The area of a triangle is calculated with the formula:

$A=\cfrac{1}{2} \, b h$ .

Area of a Right Triangle image 29 US

For this triangle:

$b = 11 \, ft$

$h = 4 \, ft$

Substitute the values into the formula.

$\begin{aligned} A & =\cfrac{1}{2} \, b h \\\\ & =\cfrac{1}{2} \, (11)(4) \\\\ & =22 \, f t^2 \end{aligned}$

8 \, cm

800 \, cm

2 \, cm

200 \, cm

The area of a triangle is calculated with the formula:

$A=\cfrac{1}{2} \, b h$ .

$\begin{aligned} & A=40 \mathrm{~cm}^2 \\\\ & b=0.1 \mathrm{~m} \\\\ & h= \, ? \end{aligned}$

Since the measurements are in different units ( $m$ and $cm$ ), they need to be converted to the same unit:

$0.1 \, m = 10 \, cm$ .

$\begin{aligned} & A=40 \mathrm{~cm}^2 \\\\ & b=10 \mathrm{~cm} \end{aligned}$

$\begin{aligned} & A=\cfrac{1}{2} \, b h \\\\ & 40=\cfrac{1}{2} \, \times 10 \times h \\\\ & 40=5 \times h \end{aligned}$

To solve, find what times $5$ is equal to $40$ .

Since $5 \times 8=40$ , the base is $8 \, cm$ .

Area of a right triangle FAQs

How do you find the perimeter of a right triangle?

The perimeter is the distance around the right triangle. To calculate perimeter, add the lengths of the three sides together.

What is the pythagorean theorem?

The pythagorean theorem is shown by the formula $a^2+b^2=c^2$ , where $c$ is the hypotenuse of a right triangle (longest side of a right triangle) and $a$ and $b$ are adjacent sides of the triangle.

The theorem explains a relationship between the sides that make up right triangles and can be used to find a missing third side of a right triangle, if the two other side lengths are given.

The next lessons are:

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