Area · Lesson 2 of 12
Area of a Triangle
“Derive the triangle area formula from rectangles, understand base-height pairs, and justify the formula for acute, right, and obtuse triangles.”
• Derive the area of a triangle from the area of a rectangle rather than memorising the formula. • Identify a valid base and its corresponding perpendicular height. • Explain why the height may lie inside, on, or outside a triangle. • Show that the formula works for acute, right, and obtuse triangles. • Use the formula to find area, base, or height.
Triangles are the basic building blocks of many polygon-area arguments. The familiar formula A = ½bh becomes much more useful when we understand why it works and what the words base and height actually mean. A side is not automatically a height, and the longest side is not automatically the best base.
We will begin with a rectangle, where area is already understood, and then connect one half of that rectangle to a triangle. From there we will extend the reasoning to triangles whose perpendicular height falls outside the drawn shape.
From a Rectangle to a Triangle
A diagonal of a rectangle divides it into two congruent triangles. Congruent figures have equal areas, and together the two triangles fill the rectangle exactly. Therefore each triangle has half the rectangle's area.
A perpendicular segment from a vertex of a triangle to the line containing the opposite side. Its length is the height corresponding to that side as base.
Choosing a Base and Its Height
Any side of a triangle may be used as the base. Once a base is chosen, however, the height is fixed by geometry: it must be measured perpendicular to that base from the opposite vertex. A sloping side that merely reaches the base is not a height unless it forms a right angle with the base line.
Problem
A triangle has base 12 cm and corresponding height 7 cm. Find its area.
- 1.Use A = ½bh.
- 2.A = ½ × 12 × 7.
- 3.A = 42 cm².
A right triangle makes the relationship especially visible. The two perpendicular sides can serve directly as a base-height pair. If the perpendicular sides are 8 cm and 5 cm, the area is half of the 8 × 5 rectangle: 20 cm².
Problem
A right triangle has perpendicular sides 9 m and 6 m. Find its area.
- 1.The two given sides meet at 90°, so they form a valid base-height pair.
- 2.A = ½ × 9 × 6 = 27.
- 3.The area is 27 m².
When the Height Falls Outside
In an obtuse triangle, choosing one side as base may place the perpendicular from the opposite vertex outside the segment. The height is still valid because it is measured to the entire line containing the base, not only to the part between the base endpoints.
One way to justify the formula is to embed the situation in larger right triangles. Suppose point D lies on the extension of base BC, so triangles ADC and ADB share the same altitude h. The desired triangle ABC can be seen as the larger triangle ADC with triangle ADB removed.
Problem
An obtuse triangle has base BC = 10 cm. The perpendicular distance from A to the line BC is 6 cm, and the foot lies outside segment BC. Find the area.
- 1.The position of the foot does not change the definition of height.
- 2.Use base 10 cm and perpendicular height 6 cm.
- 3.A = ½ × 10 × 6 = 30 cm².
Different Bases, Same Area
The same triangle can be described with three different base-height pairs. If you choose side AB as base, use the perpendicular distance from C to line AB. If you choose BC, use the distance from A to line BC. Each pair produces the same area because the physical region has not changed.
Problem
A triangle has area 36 cm². If one side of length 9 cm is chosen as base, find the corresponding height.
- 1.Start with 36 = ½ × 9 × h.
- 2.Multiply both sides by 2: 72 = 9h.
- 3.Divide by 9: h = 8 cm.
Finding a Missing Measurement
The area formula can be rearranged. When area and base are known, height equals 2A ÷ b. When area and height are known, base equals 2A ÷ h. The factor 2 appears because the original formula uses one half.
The height must be perpendicular to the selected base. In a non-right triangle, using a sloping side as h usually gives an incorrect area.
Quiz
Why is a triangle formed by a rectangle's diagonal half the rectangle's area?
What must be true of the height corresponding to a chosen base?
A triangle has base 14 cm and height 5 cm. Its area is
In an obtuse triangle, the altitude to a chosen base may
A triangle has area 48 m² and base 12 m. Its corresponding height is
Practice Problems
- Find the area of a triangle with base 18 cm and height 11 cm.
- A right triangle has perpendicular sides 7 m and 12 m. Find its area.
- An obtuse triangle has base 15 cm and perpendicular distance 8 cm to the base line. Find its area.
- A triangle has area 63 cm² and base 14 cm. Find the corresponding height.
- A triangle has area 52 m² and height 8 m. Find the corresponding base.
- Explain in words why an altitude outside an obtuse triangle is still a valid height.
- Draw one triangle and show three different base-height pairs.
Key Takeaways
• A triangle occupies half the area of a corresponding rectangle with the same base and height. • The formula A = ½bh depends on a base and its perpendicular height. • Any side may be the base, but the matching height changes with the choice. • An altitude can lie inside the triangle, on a side, or outside the triangle. • The same formula works for acute, right, and obtuse triangles. • Missing bases and heights can be found by rearranging A = ½bh.