E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Area of a Triangle

E2.5 A1About 40 minutes

Learning goals

I can explain why a triangle's area is half the area of a parallelogram (or rectangle) with the same base and height.
I can use the formula Area = base × height ÷ 2.
I can find the height of a triangle, even when it falls outside the triangle.
I can solve problems about the area of triangles.

Words to know

base of a triangle
Any side we choose to measure from.

height of a triangle
The straight distance from the base (or the base extended) to the opposite vertex, at a right angle to the base.

half
One of two equal parts. To find half, divide by 2.

Remember from the last lesson

The area of a parallelogram is base × height, because you can cut and move a triangle to turn it into a rectangle. You also know from the lesson on congruent shapes that cutting a rectangle along its diagonal makes two congruent triangles. Those two ideas together unlock the area of a triangle.

Let's start with a sandwich

Cut a square sandwich from corner to corner. You get two triangles that are exactly the same. Each triangle is half of the sandwich.

The same is true for every triangle: two congruent copies of any triangle fit together to make a parallelogram (or a rectangle). So a triangle is half of a parallelogram with the same base and height.

A square sandwich on a plate, cut corner to corner into two equal triangles. — Cut corner to corner, a square sandwich makes two congruent triangles. Each one is half the area of the whole sandwich. AI-generated image

Left: a 6 by 4 rectangle cut along its diagonal. Each triangle is half of 24, so 12 square units. Right: a triangle with base 6 and height 3, plus a congruent copy turned around, makes a parallelogram of 6 × 3 = 18. Each triangle is 9 square units.

The formula

Area of a triangle = base × height ÷ 2, or A = b × h ÷ 2

The base is any side you choose.
The height is the straight-up distance from the base to the opposite vertex. It makes a right angle with the base.
In a right triangle, the height is one of the sides.
In an obtuse triangle, the height may fall outside the triangle. Extend the base with a dashed line to measure it.

Worked example

Example 1: Half a rectangle

Find the area of the shaded triangle on the left of the picture.

The rectangle is 6 by 4, so its area is 6 × 4 = 24 square units.
The diagonal cuts it into two congruent triangles.
Using the formula: A = 6 × 4 ÷ 2 = 24 ÷ 2 = 12.

12 square units.

Worked example

Example 2: Use the formula

A triangle has a base of 10 cm and a height of 6 cm. Find its area.

Multiply base × height: 10 × 6 = 60.
Divide by 2: 60 ÷ 2 = 30.

30 cm²

An obtuse triangle with a base of 4 units. Its height of 3 units is measured outside the triangle, from the top vertex straight down to the base extended (dashed).

Worked example

Example 3: The height is outside

Find the area of the obtuse triangle in the picture.

The base is 4 units.
The top vertex is beyond the end of the base. Extend the base (dashed) and measure straight down from the vertex: the height is 3 units.
A = 4 × 3 ÷ 2 = 12 ÷ 2 = 6.

6 square units.

Worked example

Example 4: Missing height

A triangular sail has an area of 24 m² and a base of 8 m. What is its height?

A triangle is half of a parallelogram, so the matching parallelogram has area 24 × 2 = 48 m².
For the parallelogram, base × height = 48. So 8 × height = 48.
48 ÷ 8 = 6.
Check: 8 × 6 ÷ 2 = 24. ✓

The height is 6 m.

Rebuild it

If you forget the formula, don't panic. Draw the triangle, sketch a congruent copy turned around to make a parallelogram, and find half. Understanding why a formula works means you can always rebuild it. That's real confidence!

Just for fun

Riddle: What did the triangle say to the rectangle with the same base and height?

"You're twice the shape I am... well, twice the area, anyway!"

Hands-on activity

Two-triangle puzzle

You need: paper, a ruler, a pencil and scissors (with a grown-up).

Fold a sheet of paper in half. Draw any triangle on the top layer, keeping it away from the fold. Make it big.
Cut it out through both layers, so you have two congruent triangles.
Turn one triangle around and fit the two together along a matching side to make a parallelogram.
Measure the parallelogram's base and height. Find its area with base × height.
Now find the area of one triangle with base × height ÷ 2. Is it half the parallelogram?
Try fitting the triangles together in other ways. Do you always get a parallelogram?

Math talk: "Why do we divide by 2?"

Your turn

A triangle has a base of 8 cm and a height of 5 cm. What is its area?

A right triangle has sides of 6 cm and 7 cm next to the right angle. What is its area?

A triangle has a base of 10 m and a height of 9 m. What is its area?

On a grid, a triangle has a base of 4 units and a height of 6 units. What is its area?

Two congruent triangles fit together to make a parallelogram with an area of 30 cm². What is the area of each triangle?

A triangle has an area of 20 cm² and a base of 5 cm. What is its height?

More practice: warm-up

Write the formula for the area of a triangle.

Base 4 cm, height 2 cm. Area?

Base 6 m, height 6 m. Area?

A triangle is half of which shape with the same base and height?

10 × 3 ÷ 2 = ?

More practice: on your own

Base 9 cm, height 4 cm. Area?

Base 7 cm, height 3 cm. Area?

A team pennant is a triangle with a base of 30 cm and a height of 60 cm. How much felt does it cover?

A triangle on a grid has vertices at (1, 1), (7, 1) and (3, 5). What is its area?

A triangle and a parallelogram both have a base of 8 m and a height of 5 m. Find both areas.

Challenge

A triangle has vertices at (2, 1), (5, 1) and (8, 5). The top vertex is past the end of the base. Find the area.

A triangle has an area of 35 cm² and a height of 7 cm. What is its base?

The side of a shed looks like a rectangle 8 m wide and 5 m tall, with a triangular roof on top that is 8 m wide and 3 m tall. What is the total area?

Check your understanding

Quiz

What is the formula for the area of a triangle?

A triangle has base 12 cm and height 5 cm. What is its area?

Why do we divide by 2?

A right triangle has sides 8 m and 6 m next to its right angle. What is its area?

In an obtuse triangle, the height:

A triangle has area 18 cm² and base 6 cm. What is its height?

For parents and teachers

Big idea: The Ontario curriculum (E2.5) asks Grade 5 students to use the area relationships among rectangles, parallelograms and triangles to develop the formula for the area of a triangle and solve problems. Two congruent triangles always make a parallelogram, so a triangle's area is half of base × height.

Common difficulties:

Forgetting to divide by 2. Ask: "Is your answer the triangle, or the whole parallelogram?"
Using a slanted side as the height. The height must meet the base at a right angle.
Obtuse triangles. Show how to extend the base with a dashed line.

How to help: Help with scissors. Cut sandwiches or toast into triangles and talk about halves. Social-emotional learning (A1) is woven in: understanding a formula well enough to rebuild it builds confidence.