Area of Triangles

Study Sheet

Area of Triangles

Half of a rectangle

Where the Half Comes From

Concept
The Big Idea
copycopy
The height splits the rectangle in two; in each part, the unshaded piece is a copy of the shaded piece beside it.

Any triangle is exactly half of a rectangle with the same base and height — that is where the 12\dfrac{1}{2} in the formula comes from.

A triangle is half of a parallelogram: two copies of any triangle fit together into one. So its area is half of base ×\times height, where the height is measured straight up from the base, not along a slanted side. Any side can be the base — the answer comes out the same.

Tip
The formula

Area =12bh= \dfrac{1}{2}bh, where hh is measured perpendicular to the base, not along a slanted side.

Example
Base 10, height 6
106
The height is measured straight up from the base, not along a side.

Area =12(10)(6)=30= \dfrac{1}{2}(10)(6) = 30 square units.

Example
An obtuse triangle
45
The dashed base extension shows where the height lands.

A triangle has base 44, and its top corner leans so far over that the height, 55 units, falls outside the triangle. The formula does not care: area =12×4×5=10= \tfrac12 \times 4 \times 5 = 10 square units.

Side note
Where it goes wrong

Using a slanted side as the height. The height must meet the base at a right angle; in an obtuse triangle it can even fall outside the triangle.

Try it
Try it: base 9, height 4

A triangle has base 99 cm and height 44 cm. Find its area.

Show the answer

12×9×4=12×36=18\tfrac12 \times 9 \times 4 = \tfrac12 \times 36 = \mathbf{18} cm2^2.

In Plain Terms & More Examples

Concept
In plain terms

A triangle is half a rectangle with the same base and height, so its area is 12×base×height\tfrac{1}{2} \times \text{base} \times \text{height}.

Example
Base 8, height 5
85
The dashed rectangle is 8 × 5 = 40; the triangle is half of it.

Area =12(8)(5)=20= \dfrac{1}{2}(8)(5) = 20 square units.

Example
Base 12, height 3
123
A right triangle: one side is already the height.

Area =12(12)(3)=18= \dfrac{1}{2}(12)(3) = 18 square units.

Side note
Where it goes wrong

Forgetting the half. 8×5=408 \times 5 = 40 is the area of the whole rectangle around the triangle; the triangle is half of it, 2020.

Try it
Try it: base 14

Find the area of a triangle with base 1414 and height 55.

Show the answer

12×14×5=7×5=35\tfrac12 \times 14 \times 5 = 7 \times 5 = \mathbf{35} square units.

Parallelograms and Trapezoids

Concept
Cut and slide
bh
Cut the left triangle off and slide it to the right: a b × h rectangle.

A parallelogram is a rectangle with a triangle moved from one end to the other: area =bh= bh. A trapezoid is two triangles sharing the height: area =12(b1+b2)h= \tfrac12(b_1 + b_2)h.

Example
A trapezoid
1064
Bases 6 and 10, height 4.

Bases 66 and 1010, height 44: 12(6+10)×4=32\tfrac12(6 + 10)\times4 = 32.

Example
A parallelogram
956
Slanted side 6, but the height is 5.

Base 99, height 55, slanted side 66: area =9×5=45= 9 \times 5 = 45 square units. The slanted side is not used; the height is measured straight up.

Side note
Where it goes wrong

Using the slanted side as the height. For that parallelogram, 9×6=549 \times 6 = 54 is too big: tilting a shape over does not add area. Use the perpendicular height, 55.

Try it
Try it: composite shape
643
A rectangle with a triangle on top.

A shape is a 6×46\times4 rectangle with a triangle of base 66 and height 33 on top. Find the total area.

Show the answer

24+12⋅6⋅3=24+9=3324 + \tfrac12\cdot6\cdot3 = 24 + 9 = \mathbf{33}.

Word Problems

Concept
Area in stories

Area problems ask how much surface: paint, carpet, fabric, soil. Find the area in square units, then use it: divide by what one can or bag covers, or multiply by a price per square unit.

Keep track of the units at each step. Lengths in metres give an area in square metres, and "66 m2^2 per bag" divides square metres by square metres, leaving a plain number of bags.

Example
A sail
3 m5 m
A right-angled sail: base 3 m, height 5 m.

A triangular sail has base 33 m and height 55 m: area 12⋅3⋅5=7.5\tfrac12\cdot3\cdot5 = 7.5 m2^2.

Example
Soil for a garden

A triangular garden has base 1212 m and height 88 m. Area: 12×12×8=48\tfrac12 \times 12 \times 8 = 48 m2^2. One bag of soil covers 66 m2^2, so the garden needs 48÷6=848 \div 6 = 8 bags.

Side note
Where it goes wrong

Mixing units. A triangle with base 22 m and height 5050 cm does not have area 12×2×50=50\tfrac12 \times 2 \times 50 = 50 m2^2. Use one unit: 5050 cm =0.5= 0.5 m, so the area is 12×2×0.5=0.5\tfrac12 \times 2 \times 0.5 = 0.5 m2^2.

Try it
Try it: the missing height

A triangle has area 4242 cm2^2 and base 1212 cm. What is its height?

Show the answer

12⋅12⋅h=42\tfrac12\cdot12\cdot h = 42, so 6h=426h = 42 and h=7h = \mathbf{7} cm.

Triangles on the Coordinate Grid

Concept
Read the base and height off the grid

When one side of a triangle lies along a grid line, the base is a difference of coordinates along that line, and the height is how far the third corner is from it.

A side from (1,1)(1, 1) to (7,1)(7, 1) has length 7−1=67 - 1 = 6, and a corner at y=5y = 5 is 5−1=45 - 1 = 4 above that side. The grid hands you the perpendicular height. For a triangle with no side on a grid line, draw the rectangle around it and subtract the right triangles in its corners.

Example
Corners (1,1)(1, 1), (7,1)(7, 1), (4,5)(4, 5)
xy12345678123456(4, 5)
Base 6 along the grid line, height 4 straight up.

Base along y=1y = 1: 7−1=67 - 1 = 6. Height: 5−1=45 - 1 = 4. Area =12×6×4=12= \tfrac12 \times 6 \times 4 = 12 square units.

Example
Corners (0,0)(0, 0), (5,0)(5, 0), (0,6)(0, 6)

The two short sides lie on the axes, 55 and 66 long, and meet at a right angle. Area =12×5×6=15= \tfrac12 \times 5 \times 6 = 15 square units.

Example
Corners (1,1)(1, 1), (5,2)(5, 2), (2,5)(2, 5)
xy123456123456(1, 1)(5, 2)(2, 5)
The dashed 4 × 4 square minus three corner triangles.

No side lies on a grid line. Draw the rectangle from x=1x = 1 to 55 and y=1y = 1 to 55: area 4×4=164 \times 4 = 16. Three right triangles fill its corners: 12×4×1=2\tfrac12 \times 4 \times 1 = 2, 12×3×3=4.5\tfrac12 \times 3 \times 3 = 4.5 and 12×1×4=2\tfrac12 \times 1 \times 4 = 2. Subtract: 16−2−4.5−2=7.516 - 2 - 4.5 - 2 = 7.5 square units.

Side note
Where it goes wrong

Counting grid points instead of lengths. From x=1x = 1 to x=7x = 7 there are 77 grid points, but the base is 66 units long. Subtract coordinates to get lengths.

Try it
Try it: on the grid

Find the area of the triangle with corners (2,1)(2, 1), (8,1)(8, 1) and (5,6)(5, 6).

Show the answer

Base along y=1y = 1: 8−2=68 - 2 = 6. Height: 6−1=56 - 1 = 5. Area =12×6×5=15= \tfrac12 \times 6 \times 5 = \mathbf{15} square units.

Why It Works

Take a triangle with base bb and height hh and a second copy rotated 180∘180^\circ. They fit together along the matching side into a parallelogram of base bb and height hh, and a parallelogram's area is bhbh (slice off the triangle on one end and slide it to the other to make a b×hb\times h rectangle). Each copy is half of that: A=12bhA = \tfrac12bh. The argument never used which side was called the base, which is why all three choices agree.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.