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Lesson 5 of 10

Measuring Space: Perimeter and Area · Lesson 5 of 10

Area of a Triangle

“A triangle takes half the rectangle and still gets its own famous formula.”

Learning Objectives

• Derive triangle area from a rectangle or parallelogram. • Understand base and height for acute and obtuse triangles. • Prove that a median divides a triangle into equal areas. • Compare congruence with equal area. • Use area relationships in geometric reasoning.

A triangle with base b and perpendicular height h occupies exactly half the area of a parallelogram that has the same base and the same height. The perpendicular height is important because it measures the shortest distance from the base to the opposite vertex, not the length of a sloping side.

One simple way to understand this is to imagine making an identical copy of the triangle. If the two congruent triangles are placed together in the correct way, they form a parallelogram. Since the two triangles are exactly the same in shape and size, they divide the parallelogram into two equal areas. Therefore, one triangle must have half the area of the parallelogram.

The area of the parallelogram is base × height, or bh. Since the triangle is half of this area, the area of the triangle is ½ × b × h. This is why the familiar formula for the area of a triangle is ½bh.

Two congruent triangles forming a parallelogram Triangle ABC is duplicated and rotated. The two congruent triangles are joined along a common side to form a parallelogram with base b and perpendicular height h. Two congruent triangles form a parallelogram Duplicate triangle ABC, rotate the copy, and join the matching sides. STEP 1 Triangle ABC A B C h b Copy and rotate STEP 2 Join the congruent triangles Copy 1 Copy 2 h b 2 2 × Area of triangle = Area of parallelogram = b × h Therefore, the area of each triangle is ½ × b × h.
Two congruent triangles forming a parallelogram
Triangle areaLaTeX

Why the Formula Works for Any Triangle

For acute triangles the altitude falls inside the base. For obtuse triangles, an altitude may meet the extension of the base. The formula still uses the perpendicular distance from the vertex to the line containing the base.

A Median and Equal Areas

Definition
Median of a Triangle

A line segment joining a vertex to the midpoint of the opposite side.

Suppose AD is a median of triangle ABC. Then BD=DC. Triangles ABD and ACD also share the same perpendicular height from A to line BC. Equal bases and equal heights give equal areas.

Median area theoremLaTeX
Median dividing a triangle into two equal areas A scalene triangle ABC with D as the midpoint of BC. The median AD divides the triangle into triangles ABD and ACD, which have equal areas. A B C D Median AD Triangle ABD Area = ½K Triangle ACD Area = ½K D is the midpoint of BC: BD = DC Area(△ABD) = Area(△ACD) = ½ Area(△ABC)
Median dividing a triangle into equal areas
Important insight

Two triangles can have equal area without being congruent. Equal area tells us the amount of space is the same, not that the shapes are identical.

Worked Example: Area from Base and Height

Problem
A triangle has base 14 cm and height 9 cm. Find its area.

  1. 1.A=1/2×14×9.
  2. 2.A=63 cm².
Worked Example: Median

Problem
Triangle ABC has area 48 cm² and AD is a median. Find the areas of triangles ABD and ACD.

  1. 1.A median divides the triangle into two equal-area triangles.
  2. 2.Each area=48/2=24 cm².

Practice Problems

Practice Problems
  1. Find the area of a triangle with base 17 cm and height 8 cm.
  2. A triangle has area 54 cm² and base 12 cm. Find its height.
  3. A median divides a triangle of area 90 cm². Find the two smaller areas.
  4. Draw two non-congruent triangles having the same base and the same height. Explain why their areas are equal.
  5. In triangle ABC, D is midpoint of BC and P lies anywhere on AD. Compare areas of triangles ABP and ACP and explain.

Key Takeaways

Key Takeaways

• Triangle area is half base × perpendicular height. • The formula works for acute and obtuse triangles. • A median divides a triangle into two equal-area triangles. • Equal area does not imply congruence. • Shared heights and equal bases are powerful tools in area proofs.

Coming Next

Next, we find triangle area when only side lengths are known, using Heron's formula, and extend the idea to cyclic quadrilaterals.

Previous · Lesson 4

Area of Rectangles and Parallelograms

Next · Lesson 6

Heron's Formula and Brahmagupta's Formula