Why a Triangle's Area Is Half a Rectangle

The figure shows two triangles, marked 1 and 2. The horizontal axis is the $x$-axis and the vertical axis is the $y$-axis. Each grid square is $1$ unit wide and $1$ unit tall.

Together the triangles make a four-sided shape. Its corners, called vertices, are at $(1, 1)$, $(7, 1)$, $(7, 5)$ and $(1, 5)$. Two of its sides lie on horizontal grid lines and two lie on vertical grid lines. Horizontal and vertical grid lines meet at right angles, so the shape has four right angles. It is a rectangle.

The rectangle is $7 - 1 = 6$ units long and $5 - 1 = 4$ units wide, so its area is $6 \times 4 = 24$ square units.

A diagonal of a rectangle is a straight line from one vertex to the opposite vertex. The diagonal from $(1, 1)$ to $(7, 5)$ cuts this rectangle into two triangles. Triangle 1 has vertices $(1, 1)$, $(7, 1)$ and $(7, 5)$. Triangle 2 has vertices $(1, 1)$, $(7, 5)$ and $(1, 5)$. Each one keeps one of the rectangle's right angles, so each is a right triangle.

Are the two triangles the same size? Test it with a half turn. Picture triangle 1 turning halfway around the center of the rectangle. The center is the point $(4, 3)$. It is halfway between the left and right sides and halfway between the bottom and top sides.

The vertex $(1, 1)$ is $3$ units left of the center and $2$ units below it. After a half turn, it is $3$ units right of the center and $2$ units above it, at $(7, 5)$. In the same way, $(7, 1)$ moves to $(1, 5)$, and $(7, 5)$ moves to $(1, 1)$.

Those three points are exactly the vertices of triangle 2, so triangle 1 lands right on top of triangle 2. A turn does not stretch or shrink a shape, so the two triangles have equal areas.

The triangles fill the rectangle with no gap and no overlap, so their areas add up to $24$ square units. Two equal areas that add up to $24$ must each be $24 \div 2 = 12$ square units. Each right triangle has half the area of the rectangle.

Every right triangle is half of a rectangle in the same way. A half-turn copy fits against the side across from the right angle, and the two triangles make a rectangle. Two words tell you which rectangle.

• The base is one side of the triangle. You choose which side.
• The height is the distance from the base to the vertex across from it, measured along a line that is perpendicular to the base. Perpendicular means meeting at a right angle.

In a right triangle, choose one of the two sides that form the right angle as the base. The other of those two sides is perpendicular to the base, so its length is the height.

Let $b$ be the length of the base and $h$ be the height, both in the same unit of length. The rectangle that is $b$ long and $h$ wide has area $b \times h$. The triangle is half of it, so its area is $(b \times h) \div 2$. Dividing by $2$ keeps one of two equal parts. Multiplying by $\frac{1}{2}$ does the same, so $\frac{1}{2} \times b \times h$ has the same value.

A right triangle is half of a rectangle. Use the two sides that form the right angle as the base $b$ and the height $h$. Multiply them to get the rectangle's area, then take half: $A = \frac{1}{2} \times b \times h$. Before you calculate, check that the height is perpendicular to the base. Give the area in square units.

Next, you will find the area of acute and obtuse triangles, and you will find a missing base or height of a triangle.