How to find Area of a Triangle Worksheet, Formulas, and Examples

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Key Points about Finding Area of a Triangle

The area of a triangle is the measure of the region enclosed by the three sides of the triangle.
The formula for calculating the area of a triangle is simple and easy to remember.
The area of a triangle can be calculated by multiplying the base and height of the triangle and dividing the product by two.

Finding Area of a Triangle

The area of a triangle is one of the fundamental concepts in geometry and mathematics. It is the measure of the region enclosed by the three sides of a triangle. The formula for calculating the area of a triangle is simple and easy to remember, making it an essential topic for students of all levels.

A triangle is a polygon with three sides and three angles. It is a fundamental shape in geometry and often used in real-world applications. The area of a triangle is calculated by multiplying the base and height of the triangle and dividing the product by two. The formula can be easily applied to any triangle, regardless of its size or shape.

A triangle is a 2-D three sided figure that has three angles. The three angles of a triangle always add up to 180 degrees. In order to find the Area of a Triangle, you have to multiply the length of the base by the length of the height and then divide by two. Finding the Area of a Triangle follows the same formula as finding the area of a rectangle except you have to divide by two after you multiply the base and height together. Typically, you will see a dotted line in a triangle that represents the height.

Common Core Standard: 6.G.1
Related Topics: Area of a Parallelogram, Area of a Trapezoid, Area of Composite Figures
Return To: Home, 6th Grade

How to find Area of a Triangle

Triangle Definition

A triangle is a two-dimensional geometric shape that has three straight sides and three angles. It is one of the basic shapes in geometry and is commonly used in various fields, including mathematics, engineering, and architecture. Triangles are classified based on the length of their sides and the measure of their angles.

Types of Triangles

There are three types of triangles based on the length of their sides:

Equilateral Triangle: This type of triangle has three equal sides and three equal angles of 60 degrees each.
Isosceles Triangle: This type of triangle has two equal sides and two equal angles opposite to those sides.
Scalene Triangle: This type of triangle has no equal sides and no equal angles.

There are also three types of triangles based on the measure of their angles:

Acute Triangle: This type of triangle has all three angles less than 90 degrees.
Right Triangle: This type of triangle has one angle equal to 90 degrees.
Obtuse Triangle: This type of triangle has one angle greater than 90 degrees.

Properties of Triangles

Triangles have several properties that are useful in solving problems related to them. Some of the important properties of triangles are:

The sum of the angles in a triangle is always 180 degrees.
The side opposite to the largest angle is the longest side, and the side opposite to the smallest angle is the shortest side.
The sum of any two sides of a triangle is always greater than the third side.
The altitude of a triangle is a perpendicular line segment from a vertex to the opposite side or its extension. The altitude can be used to find the area of a triangle.

Understanding the definition and properties of triangles is essential in solving problems related to them, including finding the area of a triangle.

Area of a Triangle Formula

Calculating the area of a triangle is an essential skill in geometry. The formula to find the area of a triangle is simple and straightforward. It is given as:

Area of a Triangle = 1/2 * base * height

Where the base and height are the two sides of the triangle that are perpendicular to each other. The base is the length of the side that is horizontal, and the height is the length of the side that is perpendicular to the base.

To use the formula, you need to know the values of the base and height of the triangle. Once you have these values, you can plug them into the formula to find the area of the triangle.

It is important to note that the base and height of the triangle must be measured in the same units, such as centimeters or inches, before plugging them into the formula.

The formula can be used for any type of triangle, whether it is a right triangle, an acute triangle, or an obtuse triangle. The formula can also be used for any shape of triangle, whether it is scalene, isosceles, or equilateral.

In summary, the formula to find the area of a triangle is straightforward and easy to use. It involves multiplying the base and height of the triangle and dividing the result by two. By knowing the values of the base and height of the triangle, one can easily calculate the area of the triangle using this formula.

How to Find Area of a Triangle Without Height

Calculating the area of a triangle can be a bit tricky if you don’t have the height. However, there are several methods that can help you find the area of a triangle without the height.

Using Base and Height

One of the most common methods is to use the base and height of the triangle. The base is one side of the triangle, and the height is the distance from the base to the opposite vertex. To find the area of the triangle, you can use the formula:

Area = 0.5 x Base x Height

If you don’t have the height, you can use other information about the triangle to find it. For example, if you know the length of two sides and the angle between them, you can use the Law of Cosines to find the length of the third side. Then, you can use the Pythagorean Theorem to find the height.

Using Heron’s Formula

Another method to find the area of a triangle without the height is to use Heron’s formula. Heron’s formula is used to find the area of a triangle when the length of the three sides of the triangle is known.

The formula is:

Area = sqrt(s(s-a)(s-b)(s-c))

where a, b, and c are the lengths of the sides of the triangle, and s is the semiperimeter of the triangle, which is half the perimeter of the triangle.

Using Trigonometry

Trigonometry can also be used to find the area of a triangle without the height. If you know the length of two sides and the angle between them, you can use the sine function to find the height. Then, you can use the formula:

Area = 0.5 x Base x Height

to find the area of the triangle.

In conclusion, there are several methods to find the area of a triangle without the height. These methods include using the base and height, Heron’s formula, and trigonometry. By using these formulas, you can easily calculate the area of a triangle even without knowing the height.

How to find Area of a Triangle Solution

4 Easy steps to Solve Area of a Triangle Examples

Calculating the area of a triangle is a fundamental concept in geometry. Here are a few examples to help understand how to calculate the area of a triangle:

Determine the length of the base.
Determine the height of the triangle.
Multiply the base times the height and then divide by two.
Be sure that your answer uses correct units.

Example 1:

Find the area of a triangle with a base of 8 cm and a height of 12 cm.

Solution:

Using the formula for the area of a triangle, we have:

Area = (base x height) / 2

Area = (8 x 12) / 2

Area = 48 cm²

Therefore, the area of the triangle is 48 cm².

Example 2:

Find the area of a right-angled triangle with sides of 6 cm and 8 cm.

Solution:

To solve this problem, we need to use the Pythagorean theorem to find the height of the triangle:

a² + b² = c²

6² + h² = 8²

h² = 8² – 6²

h² = 64 – 36

h² = 28

h = √28

h = 5.29 cm (rounded to two decimal places)

Now, we can use the formula for the area of a triangle:

Area = (base x height) / 2

Area = (6 x 5.29) / 2

Area = 15.87 cm² (rounded to two decimal places)

Therefore, the area of the triangle is 15.87 cm².

Example 3:

Find the area of an equilateral triangle with a side length of 10 cm.

Solution:

An equilateral triangle has three equal sides and three equal angles. To find the area of an equilateral triangle, we can use the formula:

Area = (√3 / 4) x side²

Area = (√3 / 4) x 10²

Area = 25√3 cm² (exact value)

Therefore, the area of the triangle is 25√3 cm².

These examples demonstrate how to calculate the area of a triangle using different formulas and methods. By understanding these examples, one can easily solve problems related to the area of a triangle.

5 Challenging Area of a Triangle Questions

Area of a Triangle Problems

When it comes to calculating the area of a triangle, there are a variety of problems that one may encounter. Here are a few common types of area of a triangle problems:

Finding the Area of a Triangle Given the Base and Height

One common type of problem involves finding the area of a triangle given the base and height. To do this, simply multiply the base by the height and divide by 2. For example, if a triangle has a base of 6 and a height of 4, the area would be (6 x 4) / 2 = 12.

Finding the Area of a Triangle Given Three Sides

Another type of problem involves finding the area of a triangle given the lengths of all three sides. In this case, you can use Heron’s formula, which states that the area of a triangle with sides a, b, and c is:

Area = √(s(s-a)(s-b)(s-c))

Where s is the semiperimeter of the triangle, or half the perimeter, which can be calculated as:

s = (a + b + c) / 2

For example, if a triangle has sides of length 3, 4, and 5, the semiperimeter would be (3 + 4 + 5) / 2 = 6, and the area would be:

Area = √(6(6-3)(6-4)(6-5))
     = √(6 x 3 x 2 x 1)
     = √36
     = 6

Practice Problem

Here’s a practice problem to help solidify your understanding of finding the area of a triangle:

What is the area of a triangle with a base of 8 and a height of 5?

To solve this problem, simply multiply the base by the height and divide by 2:

Area = (8 x 5) / 2
     = 20

So the area of the triangle is 20 square units.

Overall, there are many different types of area of a triangle problems that you may encounter, but with a solid understanding of the formulas and techniques involved, you should be able to tackle them with ease.

How to Find Area of a Triangle FAQ

What is the formula for calculating the area of a triangle?

The formula for calculating the area of a triangle is A = 1/2 × b × h, where A is the area, b is the base of the triangle, and h is the height (a straight perpendicular line drawn from the base to the highest point of the triangle).

What are the three formulas for finding the area of a triangle?

There are three formulas for finding the area of a triangle:

Using base and height: A = 1/2 × b × h
Using two sides and the included angle: A = 1/2 × a × b × sin(C)
Using all three sides: A = sqrt(s(s-a)(s-b)(s-c)), where s is half the perimeter of the triangle, and a, b, and c are the lengths of the sides.

How do you find the area of an equilateral triangle?

To find the area of an equilateral triangle, you can use the formula A = (sqrt(3)/4) × s^2, where A is the area and s is the length of one side.

What is the area of a right triangle?

The area of a right triangle is half the product of the lengths of the two legs. If the legs have lengths a and b, and the hypotenuse has length c, then A = 1/2 × ab.

How do you calculate the area of a scalene triangle?

To calculate the area of a scalene triangle, you can use the formula A = 1/2 × b × h, where b is the length of any one side, and h is the perpendicular height from that side to the opposite vertex.

How do you find the height of a triangle given its area?

To find the height of a triangle given its area, you can use the formula h = (2A) / b, where h is the height, A is the area, and b is the length of the base.

Is the area of a triangle base times height?

Yes, the formula for the area of a triangle is A = 1/2 × b × h, where b is the base of the triangle and h is the height.

Does 1/2 base times height work for all triangles?

No, the formula 1/2 base times height only works for triangles where the height is perpendicular to the base. For other triangles, you need to use a different formula.

Area of a Triangle Worksheet Video Explanation

Watch our free video on how to find Area of Triangle. This video shows how to solve problems that are on our free Area of Triangles worksheet that you can get by submitting your email above.

Watch the free Area of a Triangle video on YouTube here: Area of a Triangle Video

Video Transcript:

This video is about how to find the area of triangle. You can get the area of a triangle worksheets used in this video for free by clicking on the link in the description below. A triangle is just a three-sided figure. When thinking in terms of finding the area of a triangle you can think of it as half of the area of a rectangle that has the same side lengths as the triangle. Area of a rectangle is equal to the length times the width, or you could also think of it as the base times the height. Area of a triangle is also equal to the base times the height except you have to divide by two. It’s kind of the same formula as a rectangle except the area of a triangle is half the size of a similar rectangle. Typically, the formula used is area of a triangle is base times the height divided by two or sometimes it’s shown as one-half base times height. It does not matter which formula you use, you will get the same answer for the area of a triangle.

In the case of this example triangle here, we have a height of 8 feet and a base of 12 feet. We know our area formula is area of a triangle is base times the height divided by two. In the case of this example we know the height is going to be eight feet and we know the base is going to be 12 feet because it’s the base of the triangle. When we substitute into our area formula, we’re going to use 12 feet for the base, that’s going to go where the base was, times eight feet for the height, eight goes where the height was and then we still divide by two. Then we are going to simplify the formula 12 times 8 first and you get 96 divided by 2 and 96 divided by 2 is 48 and our units are feet. It’s 48 feet squared. Let’s do a couple practice problems on our area of a triangle worksheet.

Jumping down to the first problem on our area of a triangle worksheet we have a triangle that has a base of 12 inches and a height of 5 inches. We already know that the formula for area of a triangle is base times the height and then divided by 2. We know our base is 12 inches because it’s the base of the triangle and we know the height is 5 inches. We are going to use our formula for a triangle and we’re going to substitute in 12 for the base. We’re going to say 12 for the base times the height which, in this case, is 5. We’re going to use 5 for the height and then divided by 2 which is in the formula 12 times 5 is 60. When we simplify, we’ll get 60 divided by 2 and then 60 divided by 2 simplifies to 30. And then our units are inches so it’s 30 inches squared.

The last problem we’re going to do on our area of a triangle worksheet is number five. This is a good example problem for finding area of a triangle because the height is on the interior of this triangle. The area formula for a triangle is base times the height and then divide by two. The base in this case is going to be 18 inches. It runs the entire length of the triangle so that’s our base and the height will be 7 inches because it’s the distance from the base to the vertex of the triangle. Now we substitute our base and our height into our formula. Base is 18 inches so that goes in for b times the height which is 7 inches so that goes in for h and then divided by 2. Then when we simplify 18 times 7 we will get 126 divided by 2 and then 126 divided by 2 is 63 and our units are inches. The answer is 63 inches squared. Try these practice problems and more on the area of triangle worksheet above.