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Area of a Triangle

4 min readdecember 13, 2021

J

Jaaziel Sandoval

J

Jaaziel Sandoval

Equation and Practice for Finding the Area of Triangle

Put in simple words, the area is how much space a 2-dimensional shape takes up. The area of this flat shape is always in square units 📏 like cm2, in2, or ft2. The area is super important because it is the foundation for future geometry-related topics like volume 📦 and circumference!

It is also important to understand the difference between area and perimeter. Perimeter refers to the distance around the shape 🔳, while area refers to the space the shape takes up ⬛. To find the area of a square you multiply the base and the height, but to find the perimeter of the square you would add all the sides together, as shown below:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-lE3jmARmIgqq.png?alt=media&token=1158d7f4-0d55-43f2-94b4-afcb6f92e752


The Equation

The equation of the area of a triangle is:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-CD6uk4hYJVmq.png?alt=media&token=d4f547bc-9013-45e6-b6fc-4264f065fa7f

You may notice that this equation is very similar to the equation for the area of a square, which is base multiplied by height. Since a triangle is half a square, you must divide the base multiplied by height by two to get the area of a triangle. However, because not every triangle is the same, it is important to understand what base and height refer to. To understand this, let's look at a few examples of specific triangles: 


Specific Triangles

General Rules

  1. In right area triangles, the base and height are interchangeable.

  2. Although the height and base are interchangeable, it is important to always identify which is which.

  3. The base is the side on the bottom.

  4. The height is the distance of a straight line from the base to the opposite corner.

Right Area Triangle

A right-angle triangle is a triangle with a 90° angle. The height of this triangle is measured from the corner of the 90° angle to the corner opposite of the base, as shown in the image below.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-17WJZ1YAC66n.png?alt=media&token=03fda5c2-9a97-4259-9c41-faddc5dde15b

Scalene Triangle 

A scalene triangle is a triangle that does not have equal sides. The height of this triangle is measured from the opposite corner of the base to the base, as shown in the image below.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-JBqFSLG4EZp7.png?alt=media&token=b468e81e-5a67-4dd8-8932-95c848c277ce

Equilateral Triangle

An equilateral triangle is a triangle that has equal sides and angles. The height of this triangle is measured from the opposite corner of the base to the center of the base as shown in the image below.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-wg9DqMpo6yIC.png?alt=media&token=66aedb58-7edb-4eaf-811d-34b081d9eaf2

Obtuse Triangle

An obtuse triangle is a triangle that has an angle bigger than 90°. The height of this triangle is measured by taking the opposite corner of the base and drawing a straight line down to where the base stops. It is important to note that in an obtuse triangle, the line of the height won’t touch the base, as shown in the image below. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-NxtNlP6hNXDk.png?alt=media&token=9a730a55-89d6-4cbc-9a50-3edb78f884d1


Examples

1. Find the area of this obtuse triangle:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-MPm7w4q04ksL.png?alt=media&token=4541c76f-394f-4ad4-9e1c-2c3f165cef8a

First, identify the height and base of the triangle 🧐. The base is 4 ft. and the height is 7.5 ft. 

6 ft. is not the height of this triangle! Although this is the length of the side, it does not identify the height ❌. Remember that in obtuse triangles, the height is measured by taking the opposite corner of the base and drawing a straight line down, until the straight line stops where the base stops. 

Once you have the base and height identified, proceed to put the numbers into the equation and solve 🖩:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-7nWON34AEfeh.png?alt=media&token=900c83b5-e909-442d-8e0a-8dd6322258ff

It’s that easy 😄! 

Example 2

Find the area of this scalene triangle:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-DSd1ERbnUgLu.png?alt=media&token=f0f13cb7-15df-4178-b872-0c3c6d6c7968

First identify the height and base of the triangle. The base is 15 cm. and the height is 26 cm. Once you have the base and height identified, you’’ just plug the numbers into the equation and solve.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-cVfSI2fLvpSe.png?alt=media&token=cf6d9ef7-a877-40b6-97bb-92aa3ca144fd

Piece of cake 🍰! 


Specific Cases

All Three Sides Are Given

What if you’re given all three sides of the triangle? Well, if it’s a right triangle it won’t matter, since one of the sides will be the same length as the height and one of the sides will be the same length as the base. 

The problem arises for the other types of triangles.  Since there isn't a 90° angle, there is no straight line that can help determine the height. 

But no worries! For these situations, we can use a special equation called Heron’s formula!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-czah6AnEYpcU.png?alt=media&token=5c8ef835-4204-43ac-973e-1e02d70d1b5e

p= half the perimeter

a= 1st side of the triangle

b= 2nd side of the triangle

c= 3rd side of the triangle

Once you identify all the components, stick them into the formula. Let’s practice!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-mtfyN3r3ywor.png?alt=media&token=2c078880-74e4-4d6f-81b7-8d2ec4d1ea73

First, you have to find out what half of the perimeter is. Remember that the perimeter is all the sides added together. Once you get that sum, divide it by 2, and that is what p will be.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ekEkbbmcihfB.png?alt=media&token=da8487be-c5c3-4a94-b0a5-ec4fe438df4a

Now that you have p, add all the other numbers into the equation, it doesn’t matter which number is which letter as long as you have all the sides present. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Z7o3cOFNW3yX.png?alt=media&token=c14c6f41-11a8-408e-87fd-7aeb220376de

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-23VFqsBkQuuf.png?alt=media&token=6fdad9b7-bf4d-4f99-a21b-c92299252d80

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-PXRJOMYHzTe6.png?alt=media&token=5c0712e1-5a78-4340-b027-4a3d9448def2

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1ouIJJQLyH7M.png?alt=media&token=653116e0-e183-4d35-aa94-bb4654815a18

Bam 💥! You did it!

Area of an Equilateral Triangle Without The Height

An equilateral triangle is a triangle that has all the same sides. The simplest way to solve this is by using the following formula: 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4pI6lS5N1UXR.png?alt=media&token=c8f48692-f7b2-48ef-9fc8-a00bddc5379a

s = the length of one side of the triangle. 

It doesn’t matter what side it is since they all measure the same. Now that you’ve identified the components, it’s time to practice:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-tJ4zklno0K3U.png?alt=media&token=fc874f63-90ae-4a32-84a3-449b369cf352

If you’re able to solve this without a calculator, props to you, but for the rest of us, just plug the equation into a calculator. It’ll look something like this:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-PMi7Rojrrm5h.png?alt=media&token=8f1016e9-e7fe-4458-9346-7ad132028671

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-olAwdHz0RXEL.png?alt=media&token=6c8f6465-b38b-4338-a602-4f8a69e0a048

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-olid6StYN7IE.png?alt=media&token=28f87183-de08-4d40-b81d-1d6998cf7363


Closing

Congratulations on learning or reviewing the basics of finding the area of a triangle 😎! There are more specific situations, but those involve trigonometric functions, you’ll learn those later! 

Area of a Triangle

4 min readdecember 13, 2021

J

Jaaziel Sandoval

J

Jaaziel Sandoval

Equation and Practice for Finding the Area of Triangle

Put in simple words, the area is how much space a 2-dimensional shape takes up. The area of this flat shape is always in square units 📏 like cm2, in2, or ft2. The area is super important because it is the foundation for future geometry-related topics like volume 📦 and circumference!

It is also important to understand the difference between area and perimeter. Perimeter refers to the distance around the shape 🔳, while area refers to the space the shape takes up ⬛. To find the area of a square you multiply the base and the height, but to find the perimeter of the square you would add all the sides together, as shown below:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-lE3jmARmIgqq.png?alt=media&token=1158d7f4-0d55-43f2-94b4-afcb6f92e752


The Equation

The equation of the area of a triangle is:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-CD6uk4hYJVmq.png?alt=media&token=d4f547bc-9013-45e6-b6fc-4264f065fa7f

You may notice that this equation is very similar to the equation for the area of a square, which is base multiplied by height. Since a triangle is half a square, you must divide the base multiplied by height by two to get the area of a triangle. However, because not every triangle is the same, it is important to understand what base and height refer to. To understand this, let's look at a few examples of specific triangles: 


Specific Triangles

General Rules

  1. In right area triangles, the base and height are interchangeable.

  2. Although the height and base are interchangeable, it is important to always identify which is which.

  3. The base is the side on the bottom.

  4. The height is the distance of a straight line from the base to the opposite corner.

Right Area Triangle

A right-angle triangle is a triangle with a 90° angle. The height of this triangle is measured from the corner of the 90° angle to the corner opposite of the base, as shown in the image below.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-17WJZ1YAC66n.png?alt=media&token=03fda5c2-9a97-4259-9c41-faddc5dde15b

Scalene Triangle 

A scalene triangle is a triangle that does not have equal sides. The height of this triangle is measured from the opposite corner of the base to the base, as shown in the image below.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-JBqFSLG4EZp7.png?alt=media&token=b468e81e-5a67-4dd8-8932-95c848c277ce

Equilateral Triangle

An equilateral triangle is a triangle that has equal sides and angles. The height of this triangle is measured from the opposite corner of the base to the center of the base as shown in the image below.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-wg9DqMpo6yIC.png?alt=media&token=66aedb58-7edb-4eaf-811d-34b081d9eaf2

Obtuse Triangle

An obtuse triangle is a triangle that has an angle bigger than 90°. The height of this triangle is measured by taking the opposite corner of the base and drawing a straight line down to where the base stops. It is important to note that in an obtuse triangle, the line of the height won’t touch the base, as shown in the image below. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-NxtNlP6hNXDk.png?alt=media&token=9a730a55-89d6-4cbc-9a50-3edb78f884d1


Examples

1. Find the area of this obtuse triangle:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-MPm7w4q04ksL.png?alt=media&token=4541c76f-394f-4ad4-9e1c-2c3f165cef8a

First, identify the height and base of the triangle 🧐. The base is 4 ft. and the height is 7.5 ft. 

6 ft. is not the height of this triangle! Although this is the length of the side, it does not identify the height ❌. Remember that in obtuse triangles, the height is measured by taking the opposite corner of the base and drawing a straight line down, until the straight line stops where the base stops. 

Once you have the base and height identified, proceed to put the numbers into the equation and solve 🖩:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-7nWON34AEfeh.png?alt=media&token=900c83b5-e909-442d-8e0a-8dd6322258ff

It’s that easy 😄! 

Example 2

Find the area of this scalene triangle:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-DSd1ERbnUgLu.png?alt=media&token=f0f13cb7-15df-4178-b872-0c3c6d6c7968

First identify the height and base of the triangle. The base is 15 cm. and the height is 26 cm. Once you have the base and height identified, you’’ just plug the numbers into the equation and solve.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-cVfSI2fLvpSe.png?alt=media&token=cf6d9ef7-a877-40b6-97bb-92aa3ca144fd

Piece of cake 🍰! 


Specific Cases

All Three Sides Are Given

What if you’re given all three sides of the triangle? Well, if it’s a right triangle it won’t matter, since one of the sides will be the same length as the height and one of the sides will be the same length as the base. 

The problem arises for the other types of triangles.  Since there isn't a 90° angle, there is no straight line that can help determine the height. 

But no worries! For these situations, we can use a special equation called Heron’s formula!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-czah6AnEYpcU.png?alt=media&token=5c8ef835-4204-43ac-973e-1e02d70d1b5e

p= half the perimeter

a= 1st side of the triangle

b= 2nd side of the triangle

c= 3rd side of the triangle

Once you identify all the components, stick them into the formula. Let’s practice!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-mtfyN3r3ywor.png?alt=media&token=2c078880-74e4-4d6f-81b7-8d2ec4d1ea73

First, you have to find out what half of the perimeter is. Remember that the perimeter is all the sides added together. Once you get that sum, divide it by 2, and that is what p will be.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ekEkbbmcihfB.png?alt=media&token=da8487be-c5c3-4a94-b0a5-ec4fe438df4a

Now that you have p, add all the other numbers into the equation, it doesn’t matter which number is which letter as long as you have all the sides present. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Z7o3cOFNW3yX.png?alt=media&token=c14c6f41-11a8-408e-87fd-7aeb220376de

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-23VFqsBkQuuf.png?alt=media&token=6fdad9b7-bf4d-4f99-a21b-c92299252d80

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-PXRJOMYHzTe6.png?alt=media&token=5c0712e1-5a78-4340-b027-4a3d9448def2

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1ouIJJQLyH7M.png?alt=media&token=653116e0-e183-4d35-aa94-bb4654815a18

Bam 💥! You did it!

Area of an Equilateral Triangle Without The Height

An equilateral triangle is a triangle that has all the same sides. The simplest way to solve this is by using the following formula: 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4pI6lS5N1UXR.png?alt=media&token=c8f48692-f7b2-48ef-9fc8-a00bddc5379a

s = the length of one side of the triangle. 

It doesn’t matter what side it is since they all measure the same. Now that you’ve identified the components, it’s time to practice:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-tJ4zklno0K3U.png?alt=media&token=fc874f63-90ae-4a32-84a3-449b369cf352

If you’re able to solve this without a calculator, props to you, but for the rest of us, just plug the equation into a calculator. It’ll look something like this:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-PMi7Rojrrm5h.png?alt=media&token=8f1016e9-e7fe-4458-9346-7ad132028671

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-olAwdHz0RXEL.png?alt=media&token=6c8f6465-b38b-4338-a602-4f8a69e0a048

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-olid6StYN7IE.png?alt=media&token=28f87183-de08-4d40-b81d-1d6998cf7363


Closing

Congratulations on learning or reviewing the basics of finding the area of a triangle 😎! There are more specific situations, but those involve trigonometric functions, you’ll learn those later! 



© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.