Area of Triangles

In this lesson, we will look at different formulas that can be used to calculate the area of triangles when we are

• given Base and Height
• given the Length of Three Sides (Heron's formula)
• given Side, Angle, Side
• given an Equilateral Triangle
• given a Triangle drawn on a Grid
• given Three Vertices on the Coordinate Plane
• given Three Vertices in a 3D space

Given base and height

If we are given the base of the triangle and the perpendicular height then we can use the formula.

Area =

Area of a triangle is equal to half of the product of its base and height.

The height of a triangle is the perpendicular distance from a vertex to the base of the triangle.

Any of the 3 sides of a triangle can be used as a base. It all depends on where the height is drawn.

Worksheet to calculate the area of triangles.

Worksheet to calculate the area of triangles and parallelograms.

Worksheet to calculate the height given the area of the triangle.

Worksheet to solve problems that involve the base, height and area of a triangle.

If you are given the sides of an isosceles or equilateral triangle, you can use the Pythagorean Theorem to find the height of the triangle and then use the above formula to find the area.

The following video shows an example of using the above formula to calculate the area of a triangle.

The following video shows how we can use the Pythagorean theorem to get the height of an isosceles triangle and then calculate the area of the triangle.

Given length of three sides

If we are given three sides of a triangle, we can use Heron’s formula:

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

Given Side Angle Side

If we are given the lengths of two sides of a triangle and the size of angle between them we can use the formula:

Area of triangle = ab sin C

The area of a triangle is equal to half the product of two sides times the sine of the included angle

By considering sin A and sin B in a similar way, we obtain

Area = bc sin A = ac sin B

We may use any of the above formulas depending on which two sides and angle are given.

Remember that the given angle must be between the two given sides.

Example:

Find the area of triangle PQR if p = 6.5 cm, r = 4.3 cm and Ð Q = 39˚. Give your answer correct to 2 decimal places.

Solution:

Area of triangle PQR

= pr sin Q

= sin 39˚ = 8.79 cm²

The following video shows how to use the above formula to find the area of an oblique triangle.

Given Equilateral Triangle

To find the area of an equilateral triangle, we can use the Pythagorean Theorem to get the height of the triangle and then use formula

Area =

or we can use the following formula:

The area of an equilateral triangle (with all sides congruent) is equal to

where s is the length of any side of the triangle

This video will show how to find the area of an equilateral triangle with side s

Given a Triangle drawn on a Grid

If the triangle is drawn on a grid then we can use the "box" method to calculate the area of the triangle.

This method involves drawing a smallest box that will enclose the triangle. Make sure that the box follows the grid of the graph paper. The space between the triangle and box is subdivided into right triangles and rectangles and the total area of the space is calculated. The area of the required triangle is then the area of the space subtracted from the area of the box.

The following video will show how to use the box method to calculate the area of a triangle drawn on a grid.

Given three vertices on the coordinate plane

When we are given three vertices of a triangle on the coordinate plane, we should first check whether the three vertices form a right triangle. If it is a right triangle then we can use the formula of half the product of its base and height to calculate the area.

If it not a right triangle then we can either use Heron's formula or the determinant of a matrix.

To use Heron's formula, we need to first get the length of each side by using the distance formula. Then plug the values into the Heron's formula

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

If you are familiar with matrices and determinants, then we can use the determinant of a matix to get the area of the triangle.

The area of a triangle is equal to

where (x₁, y₁), (x₂, y₂), (x₃, y₃) are the coordinates of the three vertices.

The plus/minus sign is meant to take whichever sign is needed to make the answer positive. If the answer is zero, then the three points are collinear (forms a straight line).

Using the determinant of a matrix we can find the area of a triangle whose coordinates are on the coordinate plane

Given three vertices in 3D space

If we are given the three vertices of a triangle in space, we can use cross products to find the area of the triangle.

If a triangle is specified by vectors u and v originating at one vertex, then the area is

The following video shows how to compute the area of a triangle given its three vertices in space, using cross products

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