In these lessons, 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 Two Vectors from One Vertex

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

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.

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.

Finding areas of triangles and parallelograms

Finding the height given the area of the triangle

Problems involving the base, height and area of a triangle

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.

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

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

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 cm^{2}

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

\(A = \frac{1}{2}bh\)or we can use the following formula:

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

\(A = \frac{{{s^2}\sqrt 3 }}{4}\)where *s* is the length of any side of the triangle

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.

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 matrix to get the area of the triangle.

The area of a triangle is equal to

\(\pm \frac{1}{2}\left| {\begin{array}{*{20}{c}}{{x_1}}&{{y_1}}&1\\{{x_2}}&{y{}_2}&1\\{{x_3}}&{{y_3}}&1\end{array}} \right|\)

where (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}), (*x*_{3}, *y*_{3}) are the coordinates of the three vertices.

Example:

Black-necked stilts are birds that live throughout Florida and surrounding areas but breed mostly in the triangular region shown on the map. Estimate the area of this region. The coordinates given are measured in miles.

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 half the magnitude of their cross product.

\(A = \frac{1}{2}\left\| {\overrightarrow u } \right. \times \left. {\overrightarrow v } \right\|\)

Example:

A triangle in R

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