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# How to find the Area of a Triangle

To find the area of a triangle, use the following formula

The area of a triangle is always half the product of the height and base.

$Area = \frac{1}{2} (base \cdot height)$

#### So which side is the base?

Any side of the triangle can be a base. All that matters is that the base and the height must be perpendicular.

Any side can be a base, but every base has only one height. The height is the line from the opposite vertex and perpendicular to the base. The illustration below shows how any leg of the triangle can be a base and the height always extends from the vertex of the opposite side and is perpendicular to the base. Play around with our applet to see how the area of a triangle can be computed from any base/height pairing.

The picture below shows you that the height can actually extend outside of the triangle. So technically the height does not necessarily intersect with the base.

### Derivation of the Area of a Triangle from Rectangle

##### Example 1

Use the formula above.

$$A = \frac{1}{2} (base \cdot height) \\ A = \frac{1}{2} (10 \cdot 3) \\ = \frac{1}{2} (30) \\ = \frac{30}{2} = 15$$

### Practice Problems

Find the area of each triangle below. Round each answer to the nearest tenth of a unit.

##### Problem 1

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (3 \cdot 3) \\ = \frac{1}{2} (9) \\ =\frac{9}{2} \\ = 4.5 \text{ inches squared}$$

##### Problem 2

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (24 \cdot 27.6) \\ = 331.2 \text{ inches squared}$$

##### Problem 3

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (12 \cdot 2.5) \\ = 15 \text{ inches squared}$$

##### Problem 4

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (12 \cdot 3.9) \\ = 23.4 \text{ inches squared}$$

##### Problem 5

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (14 \cdot 4) \\ = 28 \text{ inches squared}$$

##### Problem 6

This problems involves 1 small twist. You must decide which of the 3 bases to use. Just remember that base and height are perpendicular. Therefore, the base is '11' since it is perpendicular to the height of 13.4.

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (11 \cdot 13.4) \\ = 73.7 \text{ inches squared}$$

##### Problem 7

This problems involves 1 small twist. You must decide which of the 3 bases to use. Just remember that base and height are perpendicular. Therefore, the base is '12' since it is perpendicular to the height of 5.9.

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (12 \cdot 5.9) \\ = 35.4 \text{ inches squared}$$

##### Problem 8

Like the last problem, you must decide which of the 3 bases to use. Just remember that base and height are perpendicular. Therefore, the base is '4' since it is perpendicular to the height of 17.7.

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (4 \cdot 17.7) \\ = 35.4 \text{ inches squared}$$

##### Problem 9

Again, you must decide which of the 3 bases to use. Just remember that base and height are perpendicular. Therefore, the base is '22' since it is perpendicular to the height of 26.8.

To find the area of the triangle on the left, substitute the base and the height into the formula for area.

$$Area = \frac{1}{2} (base \cdot height) \\ =\frac{1}{2} (22 \cdot 26.8) \\ = 294.8 \text{ inches squared}$$