How to Find the Area of a Triangle

Whether you’re faced with an equilateral triangle, an isosceles triangle, or another type of triangle, you’ll need to know how to find its area.

The most basic formula is to multiply the base by the height and then divide that number by two. You can also use a formula that takes into account the sides and the included angle.

Sides

A triangle is a closed shape that has 3 sides and 3 vertices, or points where the sides meet. There are several different types of triangles in mathematics, such as right-angled, equilateral, and isosceles.

Each of these shapes has its own set of rules for finding the area of the triangle. The rules for right triangles are easy to learn, but some other triangle types may be a bit trickier to understand.

The simplest way to find the area of a triangle is to measure the lengths of the sides. This can be done with the help of a calculator or using the Pythagorean theorem.

Another way to find the area of a triangle using the sides is to use the sine rule. This rule states that the square of the length of any side of a triangle is equal to the sum of the squares of the other two sides.

Similarly, the square of any angle in a triangle is also equal to the product of the other two angles. This can be done by using the formula s(s – a)(s – b)(s – c), where s is the semiperimeter and a, b, and c are the sides of the triangle.

For a right-angled triangle, the area is half the base times the height. This is one of the most common ways to calculate the area of a triangle, but it requires knowledge of the height of the triangle and can be tricky for irregular triangles.

There are also several other formulas for calculating the area of a triangle depending on the type of triangle and the dimensions given to it. Some of these formulas are derived by the Greek mathematician Heron, and others are based on the law of cosines.

In addition to the formulas for calculating the area of equilateral, right-angled, and isosceles triangles, there are some other useful equations. The most commonly known is R=12bh, where b is the base of the triangle and h is the height.

Angles

The area of a triangle is the space that is occupied inside the three sides and angles of a triangle. This measurement is often expressed in square units and is used in different calculations for many types of figures and objects. There are a few formulas that can be used to calculate the area of triangles and these vary depending on the type of triangle.

The most common method for finding the area of a triangle is to take half the base times the height. There are other methods, however, that can be used to calculate the area of a triangle using information about the sides and angles.

You can also use trigonometric functions to find the area of a triangle. These include the inverse sine rule and the sin rule. The inverse sine rule is particularly useful when you are not sure how to solve a problem and need a quick solution.

Similarly, the sin rule can be used to solve for the area of a triangle if you are not sure how to do the base-height calculation. This is because the area of a right triangle is equal to half of its base.

One of the most helpful ways to learn how to find the area of a triangle is to use examples. This will help you to better understand how the formula works and how to apply it in real life situations.

A common example is a triangle with sides that measure a and b and an angle that is between those two sides. Once you know the lengths of a and b, you can find the area of the triangle by calculating the angle between those sides.

This formula can be used for many different types of triangles, including right triangles and equilateral triangles. There are many other formulas for triangles, too, but this is a good start.

Heron’s formula is an excellent tool to help you learn how to calculate the area of a triangle. It is a simple formula that does not use trigonometric functions directly, but it is still helpful to learn how to calculate the area of a triangular shape.

Bases

A triangle is a two-dimensional shape that has three sides and three vertices. In addition, it has one angle. A triangle has a special property called the angle sum, which means that the sum of all the interior angles in the triangle is 180 degrees.

A base, also known as the bottom, refers to any side that is perpendicular to the height or altitude of a triangle. This can be a side from an equilateral or scalene triangle or one from an isosceles triangle.

Once students understand that any side of a triangle can be a base, they may want to consider using the bases and heights from their experience with parallelograms in the previous lesson to help them find base-height pairs in different types of triangles. Explain that they can use these pairs to find the area of a triangle, since any pair of base and heights will match the formula for finding the area of a triangle: A = 1/2 x b x h (where b is the base, h is the height, and A is the area).

Ask students to work in groups of 2-4 to identify and label the bases and corresponding heights of the triangles provided. While they are working, circulate and listen to them talk about what they see. Record the phrases they use to refer to each triangle and display them for all to see during the whole-class discussion.

As the discussion progresses, select a few students to share their expressions for finding the area of any triangle with their partners and record them for all to see. As they share their reasoning, provide 1-2 minutes of quiet think time for them to begin to consider why their expression is true.

Then, call on them to restate their peers’ reasoning and describe why they feel it is correct for the triangle. This will give more students the opportunity to use language to interpret and describe their expressions, as well as to develop their reasoning more abstractly.

The height of a triangle is the distance between the base and the opposite vertex. The length of each perpendicular segment is the distance between the base and the vertex opposite it. This is a simple formula that can be used to find the height of any triangle, whether it is equilateral, scalene, or isosceles.

Height

A triangle’s height is the perpendicular line segment that is drawn from a vertex of the triangle to its opposite side. It makes a right angle with the base of the triangle that it touches, so it forms a 90 degree angle.

The height of a triangle is usually marked with the letter ‘h’ and subscripted with the name of the side it is drawn to. A triangle is equilateral when all its heights have equal lengths.

If one of the sides is shorter than the other, it is called a hypotenuse. If it is longer than the other, it is called an obtuse side.

An obtuse-angled triangle is an example of an equilateral triangle, but it has an interior angle that is greater than 90 degrees. This is an obtuse angle, and it does not form a right angle with the other sides of the triangle.

A triangle has three altitudes from each of its vertices, and they meet at a point called the orthocenter. This is the point where all the altitudes from a triangle’s vertices intersect, and the height of the triangle is the distance between that point and its opposite base.

In addition to its three altitudes, a triangle can have a fourth altitude called the median. This is an angled line segment that is drawn from one of the vertices to the other two vertices.

The area of a triangle is the product of its base and height divided by 2. This formula applies to all types of triangles, but it is particularly useful when finding the area of scalene triangles.

After presenting the different bases and heights of the triangles you have just studied, ask students to identify whether each one is a true base or a false base. For each “true” vote, record and display the statement on the wall for all to see.

For each “false” vote, discuss with students whether or not they think the statements are true. Encourage them to use examples and counterexamples to support their arguments.

For each equilateral triangle, ask student pairs to draw and explain how they understand the base of the triangles (dash segments) and the corresponding height of the triangles (lines going through a vertex). Give them 5 minutes for their pairs to work, followed by a short discussion time in the group. When the pairs have finished their explanations, collect them and display them in the classroom for everyone to see during a whole-class discussion.