△Math Explorer · Grade 5
Lesson progress0 of 10 missions
Chapter 4 · Plane Geometry

Lesson 4.1 — Choose a Base and Its Perpendicular Height

第4讲 · 选底与对应的垂直高

Area formulas become dependable when every base is paired with the correct height. You will learn to spot that pair, work backward from area, and recognize a height even when it falls outside a triangle.

Choose the base. Drop a right angle. Then use the formula.
Grade 5 enrichment 35–50 minutes Interactive SVG diagrams Autosaves in this browser
Mission 1

Meet the area formula family

Not completed

Area tells how many square units cover a region. For the figures in this chapter, the formulas all depend on a base and its perpendicular height.

In these formulas, A means area, b is the chosen base length, and h is its perpendicular height. For a trapezoid, a is the other parallel base length. Use the same length unit for all dimensions.

Triangle

A = ½ × b × h

Half of a matching parallelogram.

Parallelogram

A = b × h

A sloping side is not automatically the height.

Trapezoid

A = ½ × (a + b) × h

The bases a and b are the parallel sides.

Mission 2

One triangle can have three base–height pairs

Not completed

Any side of a triangle may be chosen as the base. Once you choose it, the height is the perpendicular segment from the opposite vertex to the line containing that base.

Triangle with a selectable base and matching height Use the buttons to highlight each side as a base and the perpendicular height from the opposite vertex. ABC base BC height from A
The area of the triangle does not change when you choose a different base. The numerical base length and height may change, but ½ × base × height stays the same.
Base BC is highlighted. Its matching height runs from A perpendicular to BC.
1
Choose one side
That side becomes the base.
2
Find the opposite vertex
It is the vertex not on the base.
3
Drop a perpendicular
Look for a 90° angle.

View all three base choices to complete this mission.

Mission 3

Use the right-angle test

Not completed

A triangle’s height runs from the vertex opposite the chosen base to the line containing that base, meeting it at 90°. Being long or vertical on the page is not enough.

Triangle with three candidate segments Only segment q is perpendicular to the horizontal base. p q r chosen base
Mission 4

Use an area formula backward

Not completed

An area formula can find area, but it can also reveal a missing base or height. Keep the base and height paired correctly, then undo the operations.

Triangle height

54 = ½ × 12 × hh = 54 × 2 ÷ 12

Parallelogram base

63 = b × 7b = 63 ÷ 7

Trapezoid height

72 = ½ × (8 + 10) × hh = 72 × 2 ÷ 18
cm
cm
cm
Mission 5

A triangle can reveal a trapezoid’s height

Not completed

The shaded triangle below has area 150 cm². Its base is the trapezoid’s 15 cm top base. Its opposite vertex lies on the other parallel base, so the triangle and trapezoid share the same perpendicular height.

Trapezoid with a shaded triangle sharing its height The top base is 15, the bottom base is 25, and the shaded triangle has area 150. 15 cm 25 cm shaded area150 cm² h

Diagram not necessarily to scale.

1
Find the shared height
150 = ½ × 15 × h
2
Use both trapezoid bases
A = ½ × (15 + 25) × h
3
State square units
The final answer is an area.
cm
cm²
Mission 6

Use the whole area as a length budget

Not completed

The entire parallelogram has area 48 cm² and perpendicular height 6 cm. The unshaded part of its top base is 5 cm. Find the area of the shaded triangle.

Parallelogram divided by an altitude into an unshaded part and a shaded triangle The parallelogram area is 48, its height is 6, and the unshaded top segment is 5. 5 cm 6 cm whole area = 48 cm² shaded full base = ?

The vertical dashed segment is perpendicular to both parallel bases.

full base = 48 ÷ 6 shaded base = full base − 5 shaded area = ½ × shaded base × 6
cm
cm
cm²
Mission 7

A height may land outside the triangle

Not completed

For an obtuse triangle, the perpendicular from the opposite vertex may meet an extension of the chosen base. It is still a valid height because it measures the distance between the vertex and the base line.

Obtuse triangle whose height lands on an extension of the base Base BC is horizontal, while the perpendicular from A meets the base line to the left of B. ABC base BC = 10 cm height = 6 cm base line extension
Do not measure the drawing. Geometry diagrams explain relationships; use only stated or logically derived lengths.
cm²
Mission 8

Base–height pair detective

Not completed

Use the formula only after you have identified the correct perpendicular height.

Case A · Triangle

Base = 12 cm; perpendicular height = 5 cm.

Case B · Parallelogram

Base = 9 cm; sloping side = 7 cm; perpendicular height = 4 cm.

Case C · Trapezoid

Parallel bases = 6 cm and 14 cm; height = 5 cm.

Case D · Definition

A segment from a triangle’s opposite vertex meets the chosen base line at 90°, but its foot lies beyond the base side.

Mission 9

Geometry workshop

Not completed

Solve at least 6 of the 8 questions correctly. Units are included in the questions, so enter only the number unless a selection box is shown.

1A triangle has base 14 cm and height 9 cm. Find its area.

2A parallelogram has area 96 cm² and height 8 cm. Find its base.

3A trapezoid has bases 12 cm and 20 cm and height 7 cm. Find its area.

4A triangle has area 45 cm² and base 10 cm. Find its height.

5In Mission 5’s trapezoid example, find the shared height in cm.

6In Mission 6’s parallelogram example, find the shaded triangle’s area in cm².

7An obtuse triangle has base 15 cm and an external perpendicular height of 8 cm. Find its area.

8A triangle’s base doubles while its perpendicular height stays the same. What happens to its area?

Need a strategy reminder?
  1. Choose the base.
  2. Identify the perpendicular height paired with that base.
  3. Select the correct figure formula.
  4. Work forward for area or backward for a missing length.
  5. Check that area answers use square units.

Read two different heights in one field

Exercise 1

The field consists of parallelogram ABCD and triangle EAB. BC = 9 m, CD = AB = 8 m. The parallelogram’s height perpendicular to BC is 7 m. EH = 6 m is perpendicular to AB. Find the field’s area.

Parallelogram and external triangle with two different perpendicular heightsBCDAEH9 m8 m6 m7 m

Use the marked dimensions and relationships; do not measure the drawing.

Hint — choose a first step

Opposite sides of a parallelogram are equal. The 6 m height belongs to base AB, not to BC.

Worked solution — compare after trying
  1. Parallelogram area: 9 × 7 = 63 m².
  2. AB = CD = 8. Triangle area: ½ × 8 × 6 = 24 m².
  3. These pieces do not overlap: 63 + 24 = 87 m².
Worked workshop solutions — open after attempting

Find the first step that differs from your own work, then try the problem again.

  1. Triangle: 14 × 9 ÷ 2 = 63.
  2. Undo base × height: 96 ÷ 8 = 12.
  3. Add the parallel bases, then halve: (12 + 20) × 7 ÷ 2 = 112.
  4. Undo the triangle formula: 45 × 2 ÷ 10 = 9.
  5. The small triangle shares the height: 150 × 2 ÷ 15 = 20.
  6. Full base = 48 ÷ 6 = 8. Shaded base = 8 − 5 = 3; shaded area = 3 × 6 ÷ 2 = 9.
  7. The external perpendicular still measures height: 15 × 8 ÷ 2 = 60.
  8. With height fixed, doubling the base doubles the product ½ × base × height.
Mission 10

Exit ticket

Not completed

Complete all five checks. The optional reflection is not automatically graded.

cm²
cm
cm²
cm²

Base–Height Navigator

This certifies that

can choose a base, identify its perpendicular height, work backward from area, and explain why a height may lie outside a triangle.

Lesson 4.1 completed

Teaching notes

The original first recovers the shared height as 20 cm, then obtains the trapezoid area 400 cm².

Mission 6 recreates the classroom practice directly below it: a parallelogram of area 48 cm² has height 6 cm, and the unshaded portion of its top base is 5 cm. The full base is 8 cm, the shaded triangle base is 3 cm, and the shaded area is 9 cm².

The multiple-base triangle, explicit right-angle test, external-height example, added numerical exercises, feedback, and exit ticket are instructional scaffolds created to make the lesson self-contained.Diagrams are explanatory and are not scale drawings.